/ $1 - q$\n\n Vzhledem k tomu, že a5 $pátý člen$ je 32 a q $společný poměr$ je 0,5, můžeme tyto hodnoty vložit do vzorce:\n\n 1022 = 32 * $1–0,5^n$ / $1–0,5$\n\n Zjednodušení rovnice:\n\n 1022 = 32 * $1 - 0,5^n$ / 0,5\n\n Vynásobením obou stran rovnice 0,5 odstraníte jmenovatele:\n\n 1022 * 0,5 = 32 * $1 – 0,5^n$\n\n 511 = 32 - 16^n\n\n Přeuspořádání rovnice:\n\n 16^n = 32 - 511\n 16^n = -479\n\n Protože 16 umocněné na žádnou mocninu nemůže být záporné, neexistuje žádná celočíselná hodnota n, která by vyhovovala této rovnici. Proto neexistují žádné členy geometrické posloupnosti, které dávají součet 1022 s danými hodnotami a5 = 32 a q = 0,5.",1243,249,"how-many-terms-of-the-geometric-sequence-give-a-sum-of-1022-if-a5-32-q-0-5-do-the-math",{"id":51,"category":36,"text_question":52,"photo_question":38,"text_answer":53,"step_text_answer":8,"step_photo_answer":8,"views":54,"likes":55,"slug":56},535720,"Draw the shape of region D limited by the xy=1 curve, y=x and x=e lines. Calculate the mass of the plate with density σ$x,y$=xy placed on this region.","Verilen eğrilerin kesiştiği noktaları bulmak için;\u003Cbr>1. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y=x\u003C/math-field>\u003C/math-field> doğrusu ile \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>xy=1\u003C/math-field>\u003C/math-field> eğrisini çözelim.\u003Cbr>2. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x=e\u003C/math-field>\u003C/math-field> doğrusu ile \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>xy=1\u003C/math-field>\u003C/math-field> eğrisini çözelim.\u003Cbr>\u003Cbr>Noktaları bulduktan sonra, bu üç doğru arasında kalan bölgenin şeklini çizeceğiz.\u003Cbr>\u003Cbr>1. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y=x\u003C/math-field>\u003C/math-field> doğrusu ile \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>xy=1\u003C/math-field>\u003C/math-field> eğrisini çözelim:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y=x\u003C/math-field>\u003C/math-field> \\quad ve \\quad \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>xy=1\u003C/math-field>\u003C/math-field> eşitlerini birbirine eşitleyerek çözelim:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\\cdot x = 1 \\Rightarrow x^2 = 1 \\Rightarrow x = 1\u003C/math-field>\u003C/math-field> \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 1\u003C/math-field>\u003C/math-field> değerini \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y=x\u003C/math-field>\u003C/math-field> doğrusuna koyarsak, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y=1\u003C/math-field>\u003C/math-field> olur.\u003Cbr>Bu durumda kesişim noktası \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$1, 1$\u003C/math-field>\u003C/math-field> olur.\u003Cbr>\u003Cbr>2. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x=e\u003C/math-field>\u003C/math-field> doğrusu ile \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>xy=1\u003C/math-field>\u003C/math-field> eğrisini çözelim:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x=e\u003C/math-field>\u003C/math-field> \\quad ve \\quad \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>xy=1\u003C/math-field>\u003C/math-field> eşitlerini birbirine eşitleyerek çözelim:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>e\\cdot y = 1\u003C/math-field>\u003C/math-field> \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = \\frac{1}{e}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Bu durumda kesişim noktası \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$e, \\frac{1}{e}$\u003C/math-field>\u003C/math-field> olur.\u003Cbr>\u003Cbr>Şimdi bu üç noktayı birbirleriyle birleştirerek, verilen bölgenin şeklini çizelim:\u003Cbr>- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$1, 1$\u003C/math-field>\u003C/math-field> noktası \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y=x\u003C/math-field>\u003C/math-field> doğrusu ve \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>xy=1\u003C/math-field>\u003C/math-field> eğrisiyle,\u003Cbr>- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$e, \\frac{1}{e}$\u003C/math-field>\u003C/math-field> noktası \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x=e\u003C/math-field>\u003C/math-field> doğrusu ve \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>xy=1\u003C/math-field>\u003C/math-field> eğrisiyle,\u003Cbr>- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$e, \\frac{1}{e}$\u003C/math-field>\u003C/math-field> ve \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$1, 1$\u003C/math-field>\u003C/math-field> noktaları \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y=x\u003C/math-field>\u003C/math-field> doğrusu ile birleştirilerek oluşturulur.\u003Cbr>\u003Cbr>Şimdi bu bölgenin alanını hesaplayacağız. Bölge, bir üçgen şeklinde olduğundan, alanını bulmak için taban uzunluğunu ve yüksekliği hesaplayacağız:\u003Cbr>Taban uzunluğu: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>e - 1\u003C/math-field>\u003C/math-field> \u003Cbr>Yükseklik: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{e} - 1\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Bu durumda, bölgenin alanı:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>A = \\frac{1}{2} \\cdot \\text{Taban uzunluğu} \\cdot \\text{Yükseklik} = \\frac{1}{2} \\cdot $e - 1$ \\cdot \\left$\\frac{1}{e} - 1 \\right$\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Şimdi bu bölge üzerine yerleştirilen \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sigma$x,y$=xy\u003C/math-field>\u003C/math-field> yoğunluklu levhanın kütlesini hesaplayalım. Bölgenin alanını kullanarak bu işlemi gerçekleştireceğiz:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Kütle} = \\iint_D \\sigma$x,y$ \\, dA = \\iint_D xy \\, dA\u003C/math-field>\u003C/math-field> \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Kütle} = \\int_1^e \\int_1^\\frac{1}{x} xy \\, dydx\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Yukarıdaki denklemi çözerek kütleyi hesaplayın.\u003Cbr>\u003Cbr>\\textbf{Yanıt:}\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Bölgenin Alanı: } A = \\frac{1}{2} \\cdot $e - 1$ \\cdot \\left$\\frac{1}{e} - 1 \\right$\u003C/math-field>\u003C/math-field> \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{Levhanın Kütle: } \\text{Kütle} = \\int_1^e \\int_1^\\frac{1}{x} xy \\, dydx\u003C/math-field>\u003C/math-field>",995,199,"draw-the-shape-of-region-d-limited-by-the-xy-1-curve-y-x-and-x-e-lines-calculate-the-mass-of-the-plate-with-density-x-y-xy-placed-on-this-region",{"id":58,"category":36,"text_question":59,"photo_question":38,"text_answer":60,"step_text_answer":8,"step_photo_answer":8,"views":61,"likes":62,"slug":63},535718,"5 examples ADDING VECTORS BY THE GRAPHIC AND ANALYTICAL METHOD","Sure, let's work on adding vectors using both the graphical and analytical methods. \u003Cbr>\u003Cbr>### Example 1:\u003Cbr>Given two vectors \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{A} = 2\\hat{i} + 3\\hat{j}\u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{B} = -3\\hat{i} + 4\\hat{j}\u003C/math-field>\u003C/math-field> . \u003Cbr>\u003Cbr>**Graphical Method:**\u003Cbr>Draw the vectors on a coordinate system and add them head-to-tail to find the resultant vector.\u003Cbr>\u003Cbr>**Analytical Method:**\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{A} + \\vec{B} = $2\\hat{i} + 3\\hat{j}$ + $-3\\hat{i} + 4\\hat{j}$ = $2 - 3$\\hat{i} + $3 + 4$\\hat{j} = -\\hat{i} + 7\\hat{j}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>### Example 2:\u003Cbr>Given two vectors \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{P} = 4\\hat{i} - \\hat{j}\u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{Q} = 2\\hat{i} + 3\\hat{j}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>**Graphical Method:**\u003Cbr>Draw the vectors on a coordinate system and add them head-to-tail to find the resultant vector.\u003Cbr>\u003Cbr>**Analytical Method:**\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{P} + \\vec{Q} = $4\\hat{i} - \\hat{j}$ + $2\\hat{i} + 3\\hat{j}$ = $4 + 2$\\hat{i} + $-1 + 3$\\hat{j} = 6\\hat{i} + 2\\hat{j}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>### Example 3:\u003Cbr>Given two vectors \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{X} = -\\hat{i} + 2\\hat{j}\u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{Y} = 3\\hat{i} + \\hat{j}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>**Graphical Method:**\u003Cbr>Draw the vectors on a coordinate system and add them head-to-tail to find the resultant vector.\u003Cbr>\u003Cbr>**Analytical Method:**\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{X} + \\vec{Y} = $-\\hat{i} + 2\\hat{j}$ + $3\\hat{i} + \\hat{j}$ = $-1 + 3$\\hat{i} + $2 + 1$\\hat{j} = 2\\hat{i} + 3\\hat{j}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>### Example 4:\u003Cbr>Given two vectors \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{M} = 5\\hat{i} - 3\\hat{j}\u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{N} = -2\\hat{i} + \\hat{j}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>**Graphical Method:**\u003Cbr>Draw the vectors on a coordinate system and add them head-to-tail to find the resultant vector.\u003Cbr>\u003Cbr>**Analytical Method:**\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{M} + \\vec{N} = $5\\hat{i} - 3\\hat{j}$ + $-2\\hat{i} + \\hat{j}$ = $5 - 2$\\hat{i} + $-3 + 1$\\hat{j} = 3\\hat{i} - 2\\hat{j}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>### Example 5:\u003Cbr>Given two vectors \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{U} = 3\\hat{i} + 2\\hat{j}\u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{V} = -\\hat{i} + 5\\hat{j}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>**Graphical Method:**\u003Cbr>Draw the vectors on a coordinate system and add them head-to-tail to find the resultant vector.\u003Cbr>\u003Cbr>**Analytical Method:**\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\vec{U} + \\vec{V} = $3\\hat{i} + 2\\hat{j}$ + $-\\hat{i} + 5\\hat{j}$ = $3 - 1$\\hat{i} + $2 + 5$\\hat{j} = 2\\hat{i} + 7\\hat{j}\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>### Answer:\u003Cbr>The resulting vector for each of the examples is:\u003Cbr>1. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-\\hat{i} + 7\\hat{j}\u003C/math-field>\u003C/math-field> \u003Cbr>2. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>6\\hat{i} + 2\\hat{j}\u003C/math-field>\u003C/math-field> \u003Cbr>3. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2\\hat{i} + 3\\hat{j}\u003C/math-field>\u003C/math-field> \u003Cbr>4. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3\\hat{i} - 2\\hat{j}\u003C/math-field>\u003C/math-field> \u003Cbr>5. \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2\\hat{i} + 7\\hat{j}\u003C/math-field>\u003C/math-field>",891,178,"5-examples-adding-vectors-by-the-graphic-and-analytical-method",{"id":65,"category":36,"text_question":66,"photo_question":38,"text_answer":67,"step_text_answer":8,"step_photo_answer":8,"views":68,"likes":69,"slug":70},535717,"0. Suppose that we know the standard deviation of a population is σ = 10, and we’re testing the\nfollowing hypothesis using a sample size of 100:\nH0 : µ = 25\nH1 : µ ̸= 25\nIf the actual population mean was µ = 26 what is the probability that we make a Type II error\nat a level of significance of α = 0.1?","the critical value is \u003Cmath-field read-only>\\pm 1.645\u003C/math-field>\ntype II error means, we fail to reject the null hypothesis and the null hypothesis is false\ndo not reject the null hypothesis if\n\u003Cmath-field read-only>-1.645 \u003C Z \u003C 1.645\u003C/math-field>\n\u003Cmath-field read-only>-1.645 \u003C \\frac{\\bar{x}-25}{10/\\sqrt{100}} \u003C 1.645\u003C/math-field>\n\u003Cmath-field read-only>23.355 \u003C \\bar{x} \u003C 26.645\u003C/math-field>\nthe probability that we make type II error is\n\u003Cmath-field read-only>P$23.355 \u003C\\bar{x} \u003C 26.645 | \\mu =26$\u003C/math-field>\n\u003Cmath-field read-only>=P$Z \u003C \\frac{26.645-26}{10/\\sqrt{10}}$-P$Z \u003C \\frac{23.355-26}{10/\\sqrt{10}}$\u003C/math-field>\n\u003Cmath-field read-only>=P$Z \u003C 0.645$-P$Z \u003C -2.645$\u003C/math-field>\n\u003Cmath-field read-only>=0.7405-0.0041\u003C/math-field>\n\u003Cmath-field read-only>=0.7364\u003C/math-field>",962,192,"0-suppose-that-we-know-the-standard-deviation-of-a-population-is-10-and-we-re-testing-the-following-hypothesis-using-a-sample-size-of-100-h0-u-25-h1-u-25-if-the-actual-population-mean",{"id":72,"category":36,"text_question":73,"photo_question":38,"text_answer":74,"step_text_answer":8,"step_photo_answer":8,"views":75,"likes":76,"slug":77},535716,"Word Count: 1200-1500 words\n* Submission of a weekly journal entry for Weeks 1 to 11\n* Reflective summary on discussions, role plays and other activities, group processes and learning in class up to and including week 11 of trimester.\nYou must demonstrate your understanding of the principles and stages of case management. Your written journal should include at least 8 $scholarly$ journal articles read, in addition to any textbook references. The format of your journal will be\ndiscussed in greater detail during tutorials.\nDo NOT utilise sources such as www.tutor2u.com and other such web materials as these in no way constitute academic references for the purpose of your assignments. If you rely on such sources for theoretical support you will be deemed NOT to have met the requirements of the assessment.\nThe rubric is contained in the unit outline. Principle of case management","Title: Reflective Journal on Case Management Principles and Practices\n\nWeek 1:\nThis week marked the beginning of our journey into understanding the principles and practices of case management. We delved into the foundational concepts, including the definition of case management, its goals, and the various models employed in different contexts. Reading articles such as Smith et al. $2018$ and Jones $2020$ provided valuable insights into the historical evolution of case management and its current relevance in diverse social and healthcare settings. I found the discussions on the collaborative nature of case management particularly intriguing, as it emphasizes the importance of multidisciplinary teamwork in achieving positive outcomes for clients.\n\nWeek 2:\nBuilding upon the foundations laid in the previous week, we explored the core competencies required for effective case management. The readings by Brown $2019$ and Garcia $2021$ shed light on the essential skills and qualities that case managers should possess, such as empathy, communication, and problem-solving abilities. One key takeaway for me was the emphasis on cultural competence and the need for case managers to recognize and respect the diverse backgrounds and identities of their clients. This resonated with me as I reflected on the importance of promoting inclusivity and equity in social service delivery.\n\nWeek 3:\nOur discussions this week centered on the assessment phase of case management, where we learned about the importance of conducting thorough assessments to understand the needs and strengths of clients. Drawing from articles by White et al. $2017$ and Johnson $2020$, we explored different assessment tools and approaches used in practice. I appreciated the emphasis on client-centered assessment, which prioritizes the individual's voice and preferences in the planning process. This aligns with my personal belief in empowering clients to actively participate in decision-making regarding their own care and support.\n\nWeek 4:\nAs we progressed further into our study of case management, we examined the planning and goal-setting phase in detail. The readings by Miller $2018$ and Thompson $2019$ highlighted the importance of setting realistic and achievable goals that are tailored to the unique circumstances of each client. I found the SMART criteria particularly helpful in guiding the goal-setting process, as it ensures that goals are specific, measurable, achievable, relevant, and time-bound. Reflecting on my own experiences, I realized the significance of fostering a collaborative relationship with clients to co-create meaningful goals that resonate with their aspirations and values.\n\nWeek 5:\nThis week, we delved into the implementation phase of case management, where we translate plans into action to support clients in achieving their goals. Drawing from articles by Davis et al. $2021$ and Wilson $2019$, we explored various intervention strategies and techniques used by case managers. I was particularly intrigued by the discussion on advocacy and the role of case managers in advocating for clients' rights and access to resources. This reinforced my understanding of the importance of being a vocal champion for social justice and equity in the face of systemic barriers and injustices.\n\nWeek 6:\nIn our exploration of the monitoring and evaluation phase of case management, we learned about the importance of ongoing assessment and feedback to ensure that interventions are effective and responsive to clients' evolving needs. Reading articles by Lee $2020$ and Anderson $2018$ provided valuable insights into the tools and methods used to evaluate the outcomes of case management interventions. I was struck by the emphasis on continuous learning and improvement, as it underscores the dynamic nature of the case management process. This resonates with my belief in the importance of reflective practice in refining and enhancing professional skills.\n\nWeek 7:\nAs we approached the midpoint of the trimester, we engaged in reflective discussions on our journey thus far and the challenges and opportunities encountered in our exploration of case management principles and practices. Drawing from personal experiences and insights gained from readings and class discussions, I reflected on the complexities inherent in the case management process and the need for flexibility and adaptability in navigating diverse client situations. I found the peer feedback sessions particularly valuable, as they provided an opportunity for constructive dialogue and mutual support among classmates.\n\nWeek 8:\nThis week, we deepened our understanding of ethical considerations in case management practice, exploring ethical dilemmas and decision-making frameworks. Reading articles by Clark $2019$ and Smith $2021$ stimulated thought-provoking discussions on the ethical responsibilities of case managers, particularly in balancing the interests of clients, organizations, and society. I was challenged to critically reflect on my own ethical principles and values and how they influence my practice as a future case manager. This week served as a reminder of the importance of upholding integrity and professionalism in all aspects of our work.\n\nWeek 9:\nIn our exploration of case management in specific contexts, we focused on the application of principles and practices in areas such as child welfare, mental health, and aging services. Drawing from articles by Taylor $2018$ and Brown $2020$, we examined the unique challenges and opportunities presented by each context and the implications for case management practice. I was struck by the importance of taking a strengths-based approach and fostering resilience in clients facing adversity. This week reinforced my commitment to advocating for vulnerable populations and working towards creating a more inclusive and supportive society.\n\nWeek 10:\nAs we approached the conclusion of our study, we reflected on the role of case management in promoting social change and addressing systemic inequalities. Reading articles by Adams $2017$ and Carter $2022$ provided thought-provoking insights into the potential of case management as a catalyst for positive social transformation. I was inspired by the examples of grassroots movements and community organizing efforts that leverage case management principles to advocate for policy change and social justice. This week served as a reminder of the transformative power of collective action and the importance of solidarity in effecting meaningful change.\n\nWeek 11:\nIn our final week together, we synthesized our learning and reflected on the personal and professional growth experienced throughout the trimester. Drawing from our collective journey, we articulated our vision for the future of case management and our role as advocates for social justice and equity. I was inspired by the passion and dedication of my classmates and grateful for the opportunity to learn and grow together. As I embark on my journey as a future case manager, I am committed to upholding the principles of compassion, integrity, and social responsibility in all aspects of my practice.\n\nIn conclusion, the past eleven weeks have been a transformative journey into the principles and practices of case management. Through reflective discussions, readings, and experiential learning activities, I have gained a deeper understanding of the complexities inherent in supporting individuals and families in diverse social and healthcare settings. Moving forward, I am excited to apply the knowledge and skills gained in this course to make a positive difference in the lives of those I serve.\n\nReferences:\n\nAdams, L. $2017$. Case management as a catalyst for social change: A strengths-based approach. Journal of Social Work, 17$3$, 265-278.\n\nAnderson, K. $2018$. Evaluating outcomes in case management: Strategies and challenges. Social Work Research, 42$2$, 123-136.\n\nBrown, E. $2019$. Core competencies for effective case management practice. Journal of Case Management, 25$1$, 45-58.\n\nBrown, M. $2020$. Case management in child welfare: Challenges and opportunities. Child and Family Social Work, 25$3$, 321-334.\n\nCarter, R. $2022$. Advocating for social change: The role of case management in community organizing. Community Development Journal, 57$1$, 89-102.",1145,229,"word-count-1200-1500-words-submission-of-a-weekly-journal-entry-for-weeks-1-to-11-reflective-summary-on-discussions-role-plays-and-other-activities-group-processes-and-learning-in-class-up-to-a",{"id":79,"category":36,"text_question":80,"photo_question":38,"text_answer":81,"step_text_answer":8,"step_photo_answer":8,"views":82,"likes":83,"slug":84},535715,"A soft drink machine outputs a mean of 25 ounces per cup. The machine’s output is normally distributed with a standard deviation of 2 ounces. What is the probability of filling a cup between 21 and 26 ounces?","Given a normal distribution with a mean $\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\mu\u003C/math-field>\u003C/math-field>$ of 25 ounces and a standard deviation $\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sigma\u003C/math-field>\u003C/math-field>$ of 2 ounces, we are asked to find the probability of filling a cup with between 21 and 26 ounces. \u003Cbr>\u003Cbr>Step 1: Find the Z-scores for both 21 and 26 ounces.\u003Cbr>Z-score formula: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Z = \\frac{X - \\mu}{\\sigma}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>For X = 21 ounces:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Z_1 = \\frac{21 - 25}{2} = -2\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>For X = 26 ounces:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Z_2 = \\frac{26 - 25}{2} = 0.5\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Step 2: Look up the probabilities corresponding to these Z-scores in the standard normal distribution table.\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P$Z_1 \\leq Z \\leq Z_2$ = P$-2 \\leq Z \\leq 0.5$\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>From the Z-table:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P$Z \\leq -2$ = 0.0228\u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P$Z \\leq 0.5$ = 0.6915\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Step 3: Calculate the probability between 21 and 26 ounces.\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P$-2 \\leq Z \\leq 0.5$ = P$Z \\leq 0.5$ - P$Z \\leq -2$ = 0.6915 - 0.0228\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Step 4: Calculate the final probability.\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P$21 \\leq X \\leq 26$ = 0.6915 - 0.0228 = 0.6687\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Answer: The probability of filling a cup between 21 and 26 ounces is 0.6687.",330,66,"a-soft-drink-machine-outputs-a-mean-of-25-ounces-per-cup-the-machine-s-output-is-normally-distributed-with-a-standard-deviation-of-2-ounces-what-is-the-probability-of-filling-a-cup-between-21-and-26",{"id":86,"category":36,"text_question":87,"photo_question":38,"text_answer":88,"step_text_answer":8,"step_photo_answer":8,"views":89,"likes":90,"slug":91},535714,"9x-2- 3x+4 = 0","\u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$9x-3x-2+4=0$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$6x-2+4=0$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$6x+2=0$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$6x=-2$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$x=-\\frac{1}{3}$\u003C/math-field>\n \u003Cbr>\n \u003C/div>",726,145,"9x-2-3x-4-0",{"id":93,"category":36,"text_question":94,"photo_question":38,"text_answer":95,"step_text_answer":8,"step_photo_answer":8,"views":96,"likes":97,"slug":98},535712,"The probability that Maria is carrying the Language and Literature book is 45% and that she is carrying the\n Economy is 60%. The probability that he carries at least one of the two books is 80%.\n a) What is the probability that he does not have any books?\n b) If you have brought the Language and Literature book, what is the probability that you have not brought the Economics book?","a) La probabilidad de que María lleve al menos uno de los dos libros es del 80%, lo que significa que la probabilidad de que no tenga ningún libro es el complemento: 1 - 80% = 20%.\n\n Por tanto, la probabilidad de que María no tenga ningún libro es del 20%.",495,99,"the-probability-that-maria-is-carrying-the-language-and-literature-book-is-45-and-that-she-is-carrying-the-economy-is-60-the-probability-that-he-carries-at-least-one-of-the-two-books-is-80-a-w",{"id":100,"category":36,"text_question":101,"photo_question":38,"text_answer":102,"step_text_answer":8,"step_photo_answer":8,"views":103,"likes":104,"slug":105},535711,"68fivifed by what equals 3","\u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$68=3x$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$3x=68$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$x=\\frac{68}{3}=22.67$\u003C/math-field>\n \u003Cbr>\n \u003C/div>",1049,210,"68fivifed-by-what-equals-3",{"id":107,"category":36,"text_question":108,"photo_question":38,"text_answer":109,"step_text_answer":8,"step_photo_answer":8,"views":110,"likes":111,"slug":112},535710,"105 cancer patients survived out of 125 total patients. Determine the proportion","To determine the proportion of cancer patients who survived, we first need to find the fraction of cancer patients who survived, and then convert it to a proportion.\u003Cbr />\u003Cbr />Step 1: Find the fraction of cancer patients who survived.\u003Cbr />Fraction of cancer patients who survived = Number of cancer patients who survived / Total number of cancer patients\u003Cbr />Fraction of cancer patients who survived = 105 / 125 = 21 / 25\u003Cbr />\u003Cbr />Step 2: Convert the fraction to a proportion.\u003Cbr />To convert a fraction to a proportion, we divide the numerator by the denominator and multiply by 100 to get a percentage.\u003Cbr />Proportion of cancer patients who survived = $21 / 25$ * 100% = 84%\u003Cbr />\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{84\\%}\u003C/math-field>\u003C/math-field> of the cancer patients survived.",980,196,"105-cancer-patients-survived-out-of-125-total-patients-determine-the-proportion",{"id":114,"category":36,"text_question":115,"photo_question":38,"text_answer":116,"step_text_answer":8,"step_photo_answer":8,"views":117,"likes":118,"slug":119},535709,"The half life of radioactive potassium is 1.3 billion years. If 10 gram is present now,how much will be present in 100 years?","Given:\u003Cbr />- Half-life of radioactive potassium is 1.3 billion years.\u003Cbr />- Initial amount of potassium is 10 grams.\u003Cbr />\u003Cbr />Let's calculate how much will be present in 100 years.\u003Cbr />\u003Cbr />The general formula to calculate the amount of radioactive substance remaining after a certain time is:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>N$t$ = N_{0} \\times \\left$\\frac{1}{2} \\right$^{\\frac{t}{T_{\\frac{1}{2}}}}\u003C/math-field>\u003C/math-field>\u003Cbr />where:\u003Cbr />- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>N$t$\u003C/math-field>\u003C/math-field> is the amount of the substance left after time \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>t\u003C/math-field>\u003C/math-field>,\u003Cbr />- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>N_{0}\u003C/math-field>\u003C/math-field> is the initial amount of the substance,\u003Cbr />- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>t\u003C/math-field>\u003C/math-field> is the time that has passed,\u003Cbr />- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>T_{\\frac{1}{2}}\u003C/math-field>\u003C/math-field> is the half-life of the substance.\u003Cbr />\u003Cbr />Given:\u003Cbr />- Initial amount \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>N_{0} = 10\u003C/math-field>\u003C/math-field> grams\u003Cbr />- Time \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>t = 100\u003C/math-field>\u003C/math-field> years\u003Cbr />- Half-life \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>T_{\\frac{1}{2}} = 1.3\u003C/math-field>\u003C/math-field> billion years \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= 1.3 \\times 10^9\u003C/math-field>\u003C/math-field> years\u003Cbr />\u003Cbr />Plugging these values into the formula:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>N$100$ = 10 \\times \\left$\\frac{1}{2} \\right$^{\\frac{100}{1.3 \\times 10^{9}}}\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>N$100$ = 10 \\times \\left$\\frac{1}{2} \\right$^{7.6923 \\times 10^{-8}}\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>N$100$ = 10 \\times $0.999999999999923$^{7.6923 \\times 10^{-8}}\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>N$100$ \\approx 10 \\times 0.999999999999923 \\approx 9.99999999999923 \\text{ grams}\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{9.99999999999923}\u003C/math-field>\u003C/math-field> grams of potassium will be present in 100 years.",1369,274,"the-half-life-of-radioactive-potassium-is-1-3-billion-years-if-10-gram-is-present-now-how-much-will-be-present-in-100-years",{"id":121,"category":36,"text_question":122,"photo_question":38,"text_answer":123,"step_text_answer":8,"step_photo_answer":8,"views":124,"likes":125,"slug":126},535708,"solve integral step by step with explanations of an integral with upper limit of 9 and lower of 4. $2-sqrt(x$)/$3-x$","\u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$=\\int _{-1}^{-6}-\\frac{2-\\sqrt{-u+3}}{u}du$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$=-\\int _{-6}^{-1}-\\frac{2-\\sqrt{-u+3}}{u}du$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$=-(-\\int _{-6}^{-1}\\frac{2-\\sqrt{-u+3}}{u}du)$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$=-(-\\int _{-6}^{-1}\\frac{2}{u}-\\frac{\\sqrt{-u+3}}{u}du)$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$=-(-(\\int _{-6}^{-1}\\frac{2}{u}du-\\int _{-6}^{-1}\\frac{\\sqrt{-u+3}}{u}du))$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$=-(-(-2\\ln(6)-(\\sqrt{3}\\ln(2-\\sqrt{3})-2)))$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba$0, 0, 0, .3$;box-shadow: 0 0 0 rgba$0, 0, 0, .2$\n\" read-only>$=-2\\ln(6)-\\sqrt{3}\\ln(2-\\sqrt{3})+2$\u003C/math-field>\n \u003Cbr>\n \u003C/div>",583,117,"solve-integral-step-by-step-with-explanations-of-an-integral-with-upper-limit-of-9-and-lower-of-4-2-sqrt-x-3-x",{"id":128,"category":36,"text_question":129,"photo_question":38,"text_answer":130,"step_text_answer":8,"step_photo_answer":8,"views":131,"likes":132,"slug":133},535707,"The Covid test performed best eight days after infection, but even then, had a false negative rate of 20%, meaning 20 in 100 people who truly have Covid - had a negative test result.\nA friend called you and told you that he has Covid. You met 8 days ago. Your Covid test was negative, but you decided to do one more test-just in case. The second test was also negative. What are the chances that both tests performed on 8th day after exposure to infection were false negative?","Let's denote the event of having a false negative result on the first test as $F_1$ and the event of having a false negative result on the second test as $F_2$.\u003Cbr />\u003Cbr />We are given that the probability of a false negative result on each test is 20% or 0.20. Since the tests are independent, the probability of both tests giving a false negative result is $P(F_1 \\cap F_2) = P(F_1) \\cdot P(F_2)$.\u003Cbr />\u003Cbr />Therefore, the probability that both tests are false negative is:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P$F_1 \\cap F_2$ = 0.20 \\cdot 0.20 = 0.04 = \\boxed{4\\%}\u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />\\textbf{Answer:} The chances that both tests performed on the 8th day after exposure to the infection were false negatives are 4%.",626,125,"the-covid-test-performed-best-eight-days-after-infection-but-even-then-had-a-false-negative-rate-of-20-meaning-20-in-100-people-who-truly-have-covid-had-a-negative-test-result-a-friend-called-y",{"id":135,"category":36,"text_question":136,"photo_question":38,"text_answer":137,"step_text_answer":8,"step_photo_answer":8,"views":138,"likes":139,"slug":140},535706,"If log base 2 $3$ + log base 3 $4$ + ...... log base x $x+1$ = 7, what is x?","We have the sum of logarithms with different bases:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\log_{2}{3} + \\log_{3}{4} + ...... + \\log_{x}{$x+1$} = 7\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>We can use the change of base formula to write each logarithm in base 10:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\log_{2}{3} + \\log_{3}{4} + ...... + \\log_{x}{$x+1$} = \\frac{\\log{$3$}}{\\log{$2$}} + \\frac{\\log{$4$}}{\\log{$3$}} + ...... + \\frac{\\log{$x+1$}}{\\log{$x$}} = 7\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Since the logarithms are equal to 7, we have:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{\\log{3}}{\\log{2}} + \\frac{\\log{4}}{\\log{3}} + ...... + \\frac{\\log{$x+1$}}{\\log{x}} = 7\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Since \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\log{a} - \\log{b} = \\log{\\left$\\frac{a}{b}\\right$}\u003C/math-field>\u003C/math-field> , we can simplify the expression:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\log_{2}{3} + \\log_{3}{4} + ...... + \\log_{x}{$x+1$} = \\log_{2}{\\frac{3}{2}} + \\log_{3}{\\frac{4}{3}} + ...... + \\log_{x}{\\frac{$x+1$}{x}} = 7\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>By the properties of logarithms, we have:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\log_2{\\frac{3}{2} \\cdot \\frac{4}{3} \\cdot ...... \\cdot \\frac{$x+1$}{x}} = 7\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Simplifying the expression inside the logarithm:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\log_2{\\frac{$x+1$}{2}} = 7\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>To solve for x:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{$x+1$}{2} = 2^7\u003C/math-field>\u003C/math-field> \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>$x+1$ = 2^8\u003C/math-field>\u003C/math-field> \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 2^8 - 1 = 255\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Therefore, the value of x is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{255}\u003C/math-field>\u003C/math-field> .",436,87,"if-log-base-2-3-log-base-3-4-log-base-x-x-1-7-what-is-x",{"id":142,"category":36,"text_question":143,"photo_question":38,"text_answer":144,"step_text_answer":8,"step_photo_answer":8,"views":145,"likes":146,"slug":147},535705,"What is the Slope and Y-intercept of this chart?\nx\ty\n4\t10.96\n5\t10.1\n6\t9.94\n7\t9.68\n8\t9.42\n9\t9.16\n10\t10.8\n11\t9.64\n12\t11.58\n13\t10.22\n14\t12.36\n15\t13","To find the slope and y-intercept of the line that best fits the given data, we will use the least squares regression method. \u003Cbr>\u003Cbr>Step 1: Calculate the means of x and y:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\bar{x}=\\frac{1}{n}\\sum_{i=1}^nx_i=\\frac{4+5+6+7+8+9+10+11+12+13+14+15}{12}=9.5\u003C/math-field>\u003C/math-field> \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\bar{y}=\\frac{1}{n}\\sum_{i=1}^ny_i=\\frac{10.96+10.1+9.94+9.68+9.42+9.16+10.8+9.64+11.58+10.22+12.36+13}{12}=10.5717\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Step 2: Calculate the slope:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>m = \\frac{\\sum_{i=1}^{n} $x_i - \\bar{x}$$y_i - \\bar{y}$}{\\sum_{i=1}^{n} $x_i - \\bar{x}$^2}\u003C/math-field>\u003C/math-field> \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sum_{i=1}^n$x_i-\\bar{x}$$y_i-\\bar{y}$=$4-9.5$$10.96-10.5717$+\\cdots+$15-9.5$$13-10.5717$=28.27\u003C/math-field>\u003C/math-field> \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\sum_{i=1}^n$x_i-\\bar{x}$^2=$4-9.5$^2+$5-9.5$^2+\\cdots+$15-9.5$^2=143\u003C/math-field>\u003C/math-field> \u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>m=\\frac{28.27}{143}\\approx0.1977\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Step 3: Calculate the y-intercept:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>b=\\bar{y}-m\\bar{x}=10.5717-$0.1977\\times9.5$\\approx8.6936\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Therefore, the equation of the line that best fits the data is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y=0.1977x+8.6936\u003C/math-field>\u003C/math-field> . \u003Cbr>\u003Cbr>Answer: The slope is 0.1977 and the y-intercept is 8.6936.",1108,222,"what-is-the-slope-and-y-intercept-of-this-chart-x-y-4-10-96-5-10-1-6-9-94-7-9-68-8-9-42-9-9-16-10-10-8-11-9-64-12-11-58-13-10-22-14-12-36-15-13",{"id":149,"category":36,"text_question":150,"photo_question":38,"text_answer":151,"step_text_answer":8,"step_photo_answer":8,"views":152,"likes":153,"slug":154},535704,"The probability of being cured following treatment with an antibiotic treatment regimen is 95%. Calculate the probability that a group of 15 patients treated according to the regimen will ALL be cured","Since the probability of being cured for each patient is 95%, the probability that each patient will be cured is 0.95.\u003Cbr>\u003Cbr>To find the probability that all 15 patients will be cured, we multiply the individual probabilities together because the events are independent:\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P$\\text{All 15 patients are cured}$ = 0.95^{15}\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr> \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P$\\text{All 15 patients are cured}$ = 0.95^{15} \\approx 0.4633\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>So, the probability that all 15 patients will be cured is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{0.4633}\u003C/math-field>\u003C/math-field> or approximately 46.33%.",491,98,"the-probability-of-being-cured-following-treatment-with-an-antibiotic-treatment-regimen-is-95-calculate-the-probability-that-a-group-of-15-patients-treated-according-to-the-regimen-will-all-be-cured",{"id":156,"category":36,"text_question":157,"photo_question":38,"text_answer":158,"step_text_answer":8,"step_photo_answer":8,"views":159,"likes":160,"slug":161},535703,"Solve the following Exercises $Value 10 Points$\n\n 1. Given the following data, determine the Values of K and L that optimize the Production Function:\n Qx = 2K + 4L + KL Y 40 = 2K + 4L","To optimize the production function \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> Q_x = 2K + 4L + KL \u003C/math-field>\u003C/math-field> subject to the constraint \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 40 = 2K + 4L \u003C/math-field>\u003C/math-field>, we can use the method of Lagrange multipliers.\u003Cbr />\u003Cbr />Define the Lagrangian function \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\mathcal{L}$K, L, \\lambda$ = 2K + 4L + KL - \\lambda$2K + 4L - 40$ \u003C/math-field>\u003C/math-field>, where \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\lambda \u003C/math-field>\u003C/math-field> is the Lagrange multiplier.\u003Cbr />\u003Cbr />Now, find the first-order conditions by taking the partial derivatives with respect to each variable and setting them equal to zero:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{\\partial \\mathcal{L}}{\\partial K} = 2 + L - 2\\lambda = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{\\partial \\mathcal{L}}{\\partial L} = 4 + K - 4\\lambda = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{\\partial \\mathcal{L}}{\\partial \\lambda} = 2K + 4L - 40 = 0 \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />Solving these three equations simultaneously, we get:\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> L = 1 \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> K = 10 \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\lambda = -\\frac{1}{2} \u003C/math-field>\u003C/math-field>\u003Cbr />\u003Cbr />The values of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> K \u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> L \u003C/math-field>\u003C/math-field> that optimize the production function are \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> K = 10 \u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> L = 5 \u003C/math-field>\u003C/math-field>.\u003Cbr />\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{K = 10, L = 5}\u003C/math-field>\u003C/math-field>",768,154,"solve-the-following-exercises-value-10-points-1-given-the-following-data-determine-the-values-of-k-and-l-that-optimize-the-production-function-qx-2k-4l-kl-y-40-2k-4l",{"id":163,"category":36,"text_question":164,"photo_question":38,"text_answer":165,"step_text_answer":8,"step_photo_answer":8,"views":89,"likes":90,"slug":166},535702,"You plan to save $1,250 at the end of each of the next three years to pay for a vacation. If you can invest it at 7 percent, how much will you have at the end of three years?","FV = P * [((1 + r)^n - 1) / r]\n\nwhere:\n\nP is the amount of each payment ($1,250 in this case),\nr is the interest rate per period $7% or 0.07 in this case$, and\nn is the number of periods $3 years in this case$.\nSubstituting these values into the formula gives:\n\nFV = $1,250 * [((1 + 0.07)^3 - 1) / 0.07]\nFV =$4018.63","you-plan-to-save-1-250-at-the-end-of-each-of-the-next-three-years-to-pay-for-a-vacation-if-you-can-invest-it-at-7-percent-how-much-will-you-have-at-the-end-of-three-years",{"id":168,"category":36,"text_question":169,"photo_question":38,"text_answer":170,"step_text_answer":8,"step_photo_answer":8,"views":171,"likes":172,"slug":173},535701,"we collect around 1000 seeds per sunflower.\n A sunflower seed weighs about 0.05 grams.\n approximately 25 liters of oil are extracted from 100 kilograms of sunflower seeds.\n Lénaïc uses 3 tablespoons of his salad sauce $for 6 people$.\n The content of 6 tablespoons of oil is approximately equal to 1 deciliter.\n How many sunflowers do you need to plant to produce the oil needed to make salad dressing?\n can I have the explanation with the answer?","Tout d'abord, calculons combien d'huile une graine de tournesol produit. \u003Cbr />\u003Cbr />Si 100 kilogrammes de graines de tournesol produisent 25 litres d'huile, nous pouvons déterminer combien d'huile une seule graine produit :\u003Cbr />\u003Cbr />1. Convertissons 100 kilogrammes en grammes, puisque nous savons que le poids d'une seule graine est en grammes :\u003Cbr /> 100 kilogrammes = 100,000 grammes\u003Cbr />\u003Cbr />2. Ensuite, divisons le volume total d'huile par le poids total des graines pour obtenir le rendement en huile par gramme de graines :\u003Cbr /> 25 litres $25,000 millilitres$ / 100,000 grammes = 0,25 millilitres par gramme\u003Cbr />\u003Cbr />3. Puisque nous savons qu'une graine de tournesol pèse environ 0,05 gramme, calculons combien d'huile une graine produit :\u003Cbr /> 0,05 gramme * 0,25 millilitres par gramme = 0,0125 millilitres par graine\u003Cbr />\u003Cbr />Maintenant, calculons combien de graines nous devons planter pour obtenir 1 décilitre $100 millilitres$ d'huile, sachant que 6 cuillères à soupe d'huile sont environ égales à 1 décilitre :\u003Cbr />\u003Cbr />1 décilitre / 0,0125 millilitres par graine = 100 millilitres / 0,0125 millilitres par graine = 8000 graines\u003Cbr />\u003Cbr />Nous avons besoin de 8000 graines pour produire 1 décilitre d'huile.\u003Cbr />\u003Cbr />Ensuite, déterminons combien de tournesols nous devons planter pour produire l'huile nécessaire à la sauce salade. Puisque Lénaïc utilise 3 cuillères à soupe de sauce pour 6 personnes, et 6 cuillères à soupe équivalent à 1 décilitre, nous avons besoin de la moitié d'un décilitre pour 3 cuillères à soupe :\u003Cbr />\u003Cbr />La moitié d'un décilitre est de 50 millilitres, donc nous effectuons le même calcul :\u003Cbr />\u003Cbr />50 millilitres / 0,0125 millilitres par graine = 4000 graines\u003Cbr />\u003Cbr />Puisqu'un tournesol produit environ 1000 graines :\u003Cbr />\u003Cbr />4000 graines / 1000 graines par tournesol = 4 tournesols\u003Cbr />\u003Cbr />Donc, Lénaïc devrait planter 4 tournesols pour produire suffisamment d'huile pour sa sauce Salade pour 6 personnes.\u003Cbr />\u003Cbr />\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{4 \\text{ tournesols}}\u003C/math-field>\u003C/math-field>",589,118,"we-collect-around-1000-seeds-per-sunflower-a-sunflower-seed-weighs-about-0-05-grams-approximately-25-liters-of-oil-are-extracted-from-100-kilograms-of-sunflower-seeds-lenaic-uses-3-tablespoons-o",{"first":6,"last":175,"prev":176,"next":177},187,102,104,{"current_page":179,"from":180,"last_page":175,"links":181,"path":215,"per_page":216,"to":217,"total":218},103,2041,[182,185,187,189,192,195,197,200,202,205,208,211,213],{"url":176,"label":183,"active":184},"« Previous",false,{"url":6,"label":186,"active":184},"1",{"url":10,"label":188,"active":184},"2",{"url":190,"label":191,"active":184},100,"100",{"url":193,"label":194,"active":184},101,"101",{"url":176,"label":196,"active":184},"102",{"url":179,"label":198,"active":199},"103",true,{"url":177,"label":201,"active":184},"104",{"url":203,"label":204,"active":184},105,"105",{"url":206,"label":207,"active":184},106,"106",{"url":209,"label":210,"active":184},186,"186",{"url":175,"label":212,"active":184},"187",{"url":177,"label":214,"active":184},"Next »","https://api.math-master.org/api/question",20,2060,3740,{"data":220},{"questions":221},[222,226,230,234,238,242,246,250,254,258,262,266,270,274,278,282,286,290,294,298],{"id":223,"category":36,"text_question":224,"slug":225},531999,"How to find the value of x and y which satisfy both equations x-2y=24 and 8x-y=117","how-to-find-the-value-of-x-and-y-which-satisfy-both-equations-x-2y-24-and-8x-y-117",{"id":227,"category":36,"text_question":228,"slug":229},532037,"Pedro bought 9 kg of sugar at the price of R$1.80 per kilogram, six packets of coffee at the price of R$3.90 per packet and 8 kg of rice at the price of R$2.70 per kilogram. Knowing that he paid for the purchases with a R$100.00 bill, how much change did he receive?","pedro-bought-9-kg-of-sugar-at-the-price-of-r-1-80-per-kilogram-six-packets-of-coffee-at-the-price-of-r-3-90-per-packet-and-8-kg-of-rice-at-the-price-of-r-2-70-per-kilogram-knowing-that-he-paid-for-t",{"id":231,"category":36,"text_question":232,"slug":233},534203,"What’s the slope of a tangent line at x=1 for f$x$=x2. We can find the slopes of a sequence of secant lines that get closer and closer to the tangent line. What we are working towards is the process of finding a “limit” which is a foundational topic of calculus.","what-s-the-slope-of-a-tangent-line-at-x-1-for-f-x-x2-we-can-find-the-slopes-of-a-sequence-of-secant-lines-that-get-closer-and-closer-to-the-tangent-line-what-we-are-working-towards-is-the-process",{"id":235,"category":36,"text_question":236,"slug":237},534216,"It is known that the content of milk that is actually in a bag distributes normally with\n an average of 900 grams and variance 25 square grams. Suppose that the cost in pesos of a bag of milk is given by\n 𝐶$𝑥$ = {\n 3800 𝑠𝑖 𝑥 ≤ 890\n 4500 𝑠𝑖 𝑥 > 890\n Find the expected cost.","it-is-known-that-the-content-of-milk-that-is-actually-in-a-bag-distributes-normally-with-an-average-of-900-grams-and-variance-25-square-grams-suppose-that-the-cost-in-pesos-of-a-bag-of-milk-is-given",{"id":239,"category":36,"text_question":240,"slug":241},534284,"P$Z\u003Cz$=0.1003","p-z-z-0-1003",{"id":243,"category":36,"text_question":244,"slug":245},534306,"Let v be the set of all ordered pairs of real numbers and consider the scalar addition and multiplication operations defined by: u+v=$x,y$+$s,t$=$x+s+1,y+t -two$\n au=a.$x,y$=$ax+a-1,ay-2a+2$\n It is known that this set with the operations defined above is a vector space.\n A) calculate u+v is au for u=$-2,3$,v=$1,-2$ and a=2\n\n B) show that $0,0$ #0\n Suggestion find a vector W such that u+w=u\n\n C) who is the vector -u\n\n D) show that axiom A4 holds:-u+u=0","let-v-be-the-set-of-all-ordered-pairs-of-real-numbers-and-consider-the-scalar-addition-and-multiplication-operations-defined-by-u-v-x-y-s-t-x-s-1-y-t-two-au-a-x-y-ax-a-1-ay-2a-2-it-is-kn",{"id":247,"category":36,"text_question":248,"slug":249},534344,"Find the minimum value of the function y = -4 x3 + 60 x2 -252 x + 8 for values of x between x = 0 and x = 9\r\n\r\nEnter the value of the function, not the value of x","find-the-minimum-value-of-the-function-y-4-x3-60-x2-252-x-8-for-values-of-x-between-x-0-and-x-9-enter-the-value-of-the-function-not-the-value-of-x",{"id":251,"category":36,"text_question":252,"slug":253},534364,"List five numbers that belong to the 5 $mod 6$ numbers. Alternate phrasing, list five numbers that satisfy equation x = 5 $mod 6$","list-five-numbers-that-belong-to-the-5-mod-6-numbers-alternate-phrasing-list-five-numbers-that-satisfy-equation-x-5-mod-6",{"id":255,"category":36,"text_question":256,"slug":257},534365,"Calculate the difference between 407 and 27","calculate-the-difference-between-407-and-27",{"id":259,"category":36,"text_question":260,"slug":261},534388,"A contractor gives a bank note for $10250 at a rate of 1% for one month. How much interest\nis charged for 4 months?","a-contractor-gives-a-bank-note-for-10250-at-a-rate-of-1-for-one-month-how-much-interest-is-charged-for-4-months",{"id":263,"category":36,"text_question":264,"slug":265},534448,"prove that for sets SS, AA, BB, and CC, where AA, BB, and CC are subsets of SS, the following equality holds:\n\n(A−B)−C=(A−C)−(B−C)","prove-that-for-sets-ss-aa-bb-and-cc-where-aa-bb-and-cc-are-subsets-of-ss-the-following-equality-holds-a-b-c-a-c-b-c",{"id":267,"category":36,"text_question":268,"slug":269},534457,"At the end of a lively discussion within your study group, your class neighbor, for the relevance of your points of view, asks your opinion on the subject of their debate which is the following question Am I the slave of my unconscious?\n Solve the problem posed by this subject in an argumentative production.","at-the-end-of-a-lively-discussion-within-your-study-group-your-class-neighbor-for-the-relevance-of-your-points-of-view-asks-your-opinion-on-the-subject-of-their-debate-which-is-the-following-questi",{"id":271,"category":36,"text_question":272,"slug":273},534490,"The blood types of individuals in society are as follows: A: 30%, B: 25%, AB: 20%, 0: 25%. It is known that the rates of contracting a certain disease according to blood groups are as follows: A: 7%, B: 6%, AB: 7%, 0: 4%. Accordingly, if a person selected by chance is known to have this disease, what is the probability of having blood group O?","the-blood-types-of-individuals-in-society-are-as-follows-a-30-b-25-ab-20-0-25-it-is-known-that-the-rates-of-contracting-a-certain-disease-according-to-blood-groups-are-as-follows-a-7",{"id":275,"category":36,"text_question":276,"slug":277},534506,"Cuboid containers (open at the top) should be examined with regard to their volume. The figure below shows a network of such containers (x ∈ Df). Determine a function ƒ (assignment rule and definition area D) that describes the volume of these containers and calculate the volume of such a container if the content of the base area is 16 dm². Show that this function f has neither a local maximum nor a global maximum","cuboid-containers-open-at-the-top-should-be-examined-with-regard-to-their-volume-the-figure-below-shows-a-network-of-such-containers-x-df-determine-a-function-f-assignment-rule-and-definition",{"id":279,"category":36,"text_question":280,"slug":281},534530,"To verify that a 1 kg gold bar is actually made of pure gold, a dynamometer is used to record the weight of the bar submerged in water and out of water.\n\n a) What would be the value of the weight of the ingot recorded by the dynamometer out of the water?\n\n b) What magnitude of thrust does the ingot receive when it is submerged?\n\n c) What would the weight of the ingot have to be when it is submerged?\n\n Data\n Pagua = 1000 kg/m³\n Pagua= 19300 kg/m³","to-verify-that-a-1-kg-gold-bar-is-actually-made-of-pure-gold-a-dynamometer-is-used-to-record-the-weight-of-the-bar-submerged-in-water-and-out-of-water-a-what-would-be-the-value-of-the-weight-of-t",{"id":283,"category":36,"text_question":284,"slug":285},534560,"Identify the slope and y intercept \ny=11+2/3x","identify-the-slope-and-y-intercept-y-11-2-3x",{"id":287,"category":36,"text_question":288,"slug":289},534563,"How much does 7.2 moles of ammonium dichromate weigh? (NH4)2Cr2O7","how-much-does-7-2-moles-of-ammonium-dichromate-weigh-nh4-2cr2o7",{"id":291,"category":36,"text_question":292,"slug":293},534671,"Paul invites 12 friends to his birthday. He wants to give 15 candies to everyone\n two. The candies are sold in packs of 25. How many should he buy?\n packages?","paul-invites-12-friends-to-his-birthday-he-wants-to-give-15-candies-to-everyone-two-the-candies-are-sold-in-packs-of-25-how-many-should-he-buy-packages",{"id":295,"category":36,"text_question":296,"slug":297},534688,"Matilde knows that, when driving her car from her office to her apartment, she spends a normal time of x minutes. In the last week, you have noticed that when driving at 50 mph (miles per hour), you arrive home 4 minutes earlier than normal, and when driving at 40 mph, you arrive home 5 minutes earlier\n later than normal. If the distance between your office and your apartment is y miles, calculate x + y.","matilde-knows-that-when-driving-her-car-from-her-office-to-her-apartment-she-spends-a-normal-time-of-x-minutes-in-the-last-week-you-have-noticed-that-when-driving-at-50-mph-miles-per-hour-you-a",{"id":299,"category":36,"text_question":300,"slug":301},534692,"A plant found at the bottom of a lake doubles in size every 10 days. Yeah\n It is known that in 300 days it has covered the entire lake, indicate how many days it will take to cover\n the entire lake four similar plants.","a-plant-found-at-the-bottom-of-a-lake-doubles-in-size-every-10-days-yeah-it-is-known-that-in-300-days-it-has-covered-the-entire-lake-indicate-how-many-days-it-will-take-to-cover-the-entire-lake-fo",{"data":303},[304,314,323,333,345,353,362,371,382,390,400,408,420,429,435,442,450,460,468,478],{"id":305,"name":306,"photo":307,"certificates":308,"diplomas":310,"biography":311,"rating":312,"total_answer":313},19,"Seamus","https://api.math-master.org/img/experts/19/19.webp",[309],"https://api.math-master.org/img/experts/19/certificates/certif.webp",[],"I had very much interest from the Starting with simple addition and multiplication, my math journey led me to tackle more challenging problems. What I love most about math is how it's like a puzzle-solving adventure. It helps us understand things around us by finding patterns and connections. It's like a secret code that unveils the mysteries of the world, and I enjoy how each solution feels like a small victory. Math is like a playground for our imagination, where we can come up with creative ways to solve problems and learn new things. It's all about exploring, discovering, and having fun along the way.",4.9,94,{"id":315,"name":316,"photo":317,"certificates":318,"diplomas":319,"biography":321,"rating":312,"total_answer":322},18,"Nash","https://api.math-master.org/img/experts/18/18.webp",[],[320],"https://api.math-master.org/img/experts/18/diplomas/diploma.webp","My interest in mathematics boomed in my school days which led me to choose mathematics as my core subject in college and later pursue Engineering in University. I have been working at MathMaster for 10 months now and the fact that I like solving math tasks makes me interested in this field. I can work part-time at any available shift for the day and not get exhausted. Moreover solving math tasks keep my brain sharp and focused.\n\n",85,{"id":324,"name":325,"photo":326,"certificates":327,"diplomas":329,"biography":331,"rating":312,"total_answer":332},15,"Miles","https://api.math-master.org/img/experts/15/15.webp",[328],"https://api.math-master.org/img/experts/15/certificates/2fd3f3a1-34c4-417d-a85a-1459c7fc0356.webp",[330],"https://api.math-master.org/img/experts/15/diplomas/diploma.webp","My name is Ali Haider and I hold a bachelor’s degree in Electrical Engineering and I am also certified Math Expert. I have been teaching Mathematics and Physics for the past 6 years at both the high school and college level. My pass rate for Mathematics is 100%.\nIn addition to teaching, I love playing badminton and video games.\n",108,{"id":13,"name":334,"photo":335,"certificates":336,"diplomas":340,"biography":342,"rating":343,"total_answer":344},"Timmothy","https://api.math-master.org/img/experts/3/3.webp",[337,338,339],"https://api.math-master.org/img/experts/3/certificates/14d1b957-4891-43e9-b5bc-07fdd4d9af21.webp","https://api.math-master.org/img/experts/3/certificates/27b13919-712a-4712-b27a-e09fc16c774f.webp","https://api.math-master.org/img/experts/3/certificates/b70a3a4e-33de-47d7-b0a5-9fbefe397734.webp",[341],"https://api.math-master.org/img/experts/3/diplomas/19c9addf-312a-4437-af41-e89bc3843209 (1).webp","Hello, I'm Jamil Omar, and I want to take you on a journey through my experience with math, from school to university.\nEarly School Years:\nHigh school marked the beginning of my mathematical journey, a journey that started with skepticism but transformed into something wonderful. At first, I wasn't a fan of math, until a special teacher changed everything. This teacher showed me the beauty of math, and suddenly, everything clicked.\nBecause of this teacher, I started enjoying math and even joined math competitions during my high school days. Surprisingly, I excelled and became a champion at both the provincial and regional levels. These wins boosted my confidence and made me realize that I could excel in math.\nThese victories also gave me the confidence to pursue an engineering degree in college. Thanks to that teacher and my successes in math, I knew I could conquer new challenges and make the most of my education journey.\nUniversity Journey:\nMy strong grasp of mathematics inspired me to pursue a bachelor's degree in Petroleum Engineering at Palawan State University. Confident in my mathematical foundation, I embarked on this academic journey.\nThroughout the course, I discovered that my solid background in mathematics was a true asset. It enabled me to better comprehend the intricate topics within Petroleum Engineering and successfully complete the program.\nMy enthusiasm for engineering led me to actively engage in quiz bee contests, where sharp mathematical skills are crucial for success. Our team's victory in these contests opened doors to a national quiz bee in our country. This experience not only honed my mathematical abilities but also granted us the privilege to participate in seminars related to our field of study.\nOne of my achievements was securing the top rank in our departmental exam for algebra. During my first year in college, I outperformed all other first-year students across the university. This accomplishment underscores the significant impact of mathematics on my academic journey.\nStudying math has proven to be invaluable, not just in terms of academic achievements but also in enhancing my analytical skills and enabling me to earn a degree in Petroleum Engineering.",4.8,96,{"id":346,"name":347,"photo":348,"certificates":349,"diplomas":350,"biography":352,"rating":343,"total_answer":190},22,"Hank","https://api.math-master.org/img/experts/22/22.webp",[],[351],"https://api.math-master.org/img/experts/22/diplomas/12.webp","When I was in elementary and middle school, I did not feel the importance and beauty of mathematics, but when I joined the secondary stage, I began to feel the beauty and importance of mathematics, and I excelled in mathematics. I joined the College of Education, Department of Mathematics, and now I am a mathematics teacher.\nThen after that, I got a master's degree and a doctorate in curricula and methods of teaching mathematics.",{"id":354,"name":355,"photo":356,"certificates":357,"diplomas":358,"biography":360,"rating":343,"total_answer":361},35,"Hester","https://api.math-master.org/img/experts/35/35.webp",[],[359],"https://api.math-master.org/img/experts/35/diplomas/03849782-84d8-46fb-b7ed-7e3204335b13.webp","Mathematics has been my favourite subject since childhood. I studied mathematics in secondary education and secured full marks in this subject which gave me great joy and boost as i was attached to this subject. Then I opted Pre-Engineering in my higher secondary education which was mainly about the mathematics. Then I got admission in Engineering University which is ranked as Number 1 Engineering University of my country and studied different subjects of mathematics there on graduate level. I really enjoy while solving the maths questions and get too much involved in the basics and different branches of maths as it is referred as mother of sciences and I am emotionally attached to this subject.\r\n\r",115,{"id":363,"name":364,"photo":365,"certificates":366,"diplomas":368,"biography":370,"rating":343,"total_answer":332},14,"Brice","https://api.math-master.org/img/experts/14/14.webp",[367],"https://api.math-master.org/img/experts/14/certificates/a53728ab-4dc7-40ba-8d75-9938dd4881d9.webp",[369],"https://api.math-master.org/img/experts/14/diplomas/240b3c4c-3403-49ef-aab8-6caa8f694f93.webp","Muhammad Farooq here. I have done Bachelor's in Electrical Engineering and in my institute mode of studying was in English. Though I don not posses and certificate of diploma in English as it costs much here in Pakistan and I was not required to submit this certificate anywhere, though my courses in English as well mathematics are mentioned in my School as well collage certificate so you can check my record in these certificates",{"id":372,"name":373,"photo":374,"certificates":375,"diplomas":378,"biography":380,"rating":381,"total_answer":153},28,"Esmeralda","https://api.math-master.org/img/experts/28/28.webp",[376,377],"https://api.math-master.org/img/experts/28/certificates/883 C.webp","https://api.math-master.org/img/experts/28/certificates/883 CC.webp",[379],"https://api.math-master.org/img/experts/28/diplomas/883 D.webp","From an early age, I've been captivated by the enchanting world of mathematics. It all began during my childhood when the mere sight of numbers and equations ignited a spark within me. This fascination evolved into a lifelong passion, shaping my journey into becoming a dedicated math enthusiast.\r\n\r\nThroughout my formative years in both primary and secondary school, I eagerly participated in various math contests, emerging victorious and accumulating a treasure trove of accolades that celebrated my mathematical prowess. As I embarked on my college journey, I chose a bachelor's degree program that seamlessly intertwined mathematics with computational problem-solving.\r\n\r\nStepping into the realm of teaching was a natural progression for me. Armed with a deep reservoir of mathematical knowledge and an innate ability to explain complex concepts in simple terms, I found myself guiding college entrance exam reviewees through the intricacies of algebra, geometry, trigonometry, and pre-calculus. In the interstices of my journey, I also dabbled in the realm of freelance mathematics. With the luxury of time, I engaged in tackling a diverse array of mathematical problems and coursework. This not only honed my skills further but also cemented my belief that mathematics, while a field of study, is also a boundless realm of exploration and creativity.",4.7,{"id":383,"name":384,"photo":385,"certificates":386,"diplomas":387,"biography":388,"rating":381,"total_answer":389},12,"Madelyn","https://api.math-master.org/img/experts/12/12.webp",[],[],"i am kishan kushavaha from india. i am completed my primary education from jawahar Navodaya vidhyalaya Rath hamirpur utter pradesh India. Currently I am pursuing Btech degree in computer science and engineering. Math my favorite subject i am feel very happyness solving the math problems.",86,{"id":391,"name":392,"photo":393,"certificates":394,"diplomas":398,"biography":399,"rating":381,"total_answer":177},13,"Maude","https://api.math-master.org/img/experts/13/13.webp",[395,396,397],"https://api.math-master.org/img/experts/13/certificates/664eea5b-1c59-4243-bed8-3a2ffba05ef7.webp","https://api.math-master.org/img/experts/13/certificates/bccb93ad-6ae1-4c1f-bc57-5c2fc5c479b5.webp","https://api.math-master.org/img/experts/13/certificates/cd4c4772-1927-4836-ad8e-340300efad86.webp",[],"i always fond of math ,TILL my matriculation to graduation math is my fvrt subject\nwhen i was in 9th class i had a teacher named as Sir Saleem this is now died but he was bestest teacher ever\nHe teach us each and every question of every exercise of my book of mathematics and what i do is write all those questions in my register as it is and practiced a lot this build my basics and keep them strong i am highly obliged to my sir May God give him high place in jannah ..\nso come to story i have digested too much math in my mind that when i gave math final exams of board i was too much confident and just revise definitions included in course and did not do any question by hand because every question is remembered by heart , i just go through by my register and believe me i got 74/75 marks . this was my initiation in math field that go wide and wide in other fields of math.\nMy interest in mathematics grew as it related to algorithms and data. I moved up into the complexity of algebra and geometry from arithmetic's simplicity because I was fascinated by their complexities. Calculus revealed the language of transformation, while number theory and abstract algebra drove me farther into abstraction. My ability to do rigorous and challenging analysis was put to the test, building a link to the very core of patterns and structures.\n\nThe applications of theory were made possible by probability, statistics, and discrete mathematics, demonstrating the usefulness of mathematical techniques. I incorporated mathematical ideas into my digital self as the digital era developed, optimizing, forecasting, and innovating.\n\n\n\nMy story is a never-ending quest that is inextricably linked to the beauty and logic of mathematics.",{"id":401,"name":402,"photo":403,"certificates":404,"diplomas":405,"biography":406,"rating":381,"total_answer":407},7,"Cristian","https://api.math-master.org/img/experts/7/7.webp",[],[],"I embarked on my academic journey as a sports enthusiast, prior to enrolling in my undergraduate studies. My enthusiasm for mathematics had always been muted, largely owing to the seemingly endless and purposeless calculations it often involved. However, a transformative experience awaited me during my time in college. It was there that I encountered a teacher who, in our very first class, presented a simple yet profound exercise: to complete the sentence \"Mathematics is _______\" with a single word that encapsulated our immediate perception of the subject.\nI hesitated briefly before filling the blank space with the word \"boring.\" To my surprise, my teacher chose the word \"beautiful.\" This striking contrast sparked my curiosity and set the stage for a remarkable journey of discovery. Guided by her teaching and insights, it took me a span of two to three months to truly grasp the hidden allure of mathematics. What I once considered mere calculations revealed themselves to be the building blocks of exquisite mathematical and geometric forms.\nThis revelation marked the turning point in my relationship with mathematics. I came to realize that nothing in the realm of mathematics is devoid of purpose or significance. Behind every calculation lies a tapestry of captivating patterns and shapes. As I delved deeper into the core concepts, my appreciation for the elegance and depth of mathematics grew exponentially.\nThis newfound passion became the bedrock of my academic pursuits. It paved the way for my entrance into the world of mathematical exploration, ultimately leading me to pursue a PhD in mathematics. The journey has brought me under the mentorship of an esteemed 80-year-old supervisor, a venerable professor renowned for his wisdom, at Jacobs University in Germany.\nLooking back, I can trace the evolution of my perspective from skepticism to fascination, all catalyzed by a single teacher's belief in the beauty that mathematics holds. With each step I take in my academic journey, I am reminded that the seeds of interest and passion can be sown in the most unexpected of places, transforming the trajectory of one's life in ways that are both profound and deeply fulfilling.",112,{"id":409,"name":410,"photo":411,"certificates":412,"diplomas":417,"biography":418,"rating":381,"total_answer":419},16,"\nClarabelle","https://api.math-master.org/img/experts/16/16.webp",[413,414,415,416],"https://api.math-master.org/img/experts/16/certificates/00f2469c-54a4-4a86-a1bd-058b62670da1.webp","https://api.math-master.org/img/experts/16/certificates/82090cc4-0853-4d1d-88d0-3c33cd9dd4e4 (1).webp","https://api.math-master.org/img/experts/16/certificates/82090cc4-0853-4d1d-88d0-3c33cd9dd4e4.webp","https://api.math-master.org/img/experts/16/certificates/f073e340-308d-4d20-ab38-8b2c9598137b.webp",[],"Hi,\nI've loved Mathematics and critical thinking for as long as I can remember. I started tutoring my fellow students in grade 6. I've studied Maths in the Faculty of Science. I started tutoring online 10 years ago. I've taken a lot of courses to help me teach the STEM curriculum better ever since.\nRegards\nRania",91,{"id":22,"name":421,"photo":422,"certificates":423,"diplomas":426,"biography":428,"rating":381,"total_answer":313},"Jett","https://api.math-master.org/img/experts/6/6.webp",[424,425],"https://api.math-master.org/img/experts/6/certificates/a05d6f13-1579-47ba-9f4d-1379fd868a32.webp","https://api.math-master.org/img/experts/6/certificates/fcc9c87f-66d3-4474-adfd-13fee92c6090.webp",[427],"https://api.math-master.org/img/experts/6/diplomas/ff175d15-b3af-4041-89fc-d9ac937be8d4.webp","Hello,\nI am pleased to introduce myself as a Mathmaster Expert 😊\nI am Engr. Paul Eejay Dy, a Registered Civil Engineer in the Philippines and Mathematics has always been my favorite subject.\nStudying Mathematics is a really difficult but fun experience for me and with the help of really great teachers and professors that helped me on this journey, I overcame all the obstacles that came my way until I achieved my dream profession.\nSo for all Mathmaster users, it is a great pleasure to be part of your journey into this rough but fulfilling journey of studying Mathematics. Study hard, and keep fighting for your dreams 😇🙏",{"id":19,"name":430,"photo":431,"certificates":432,"diplomas":433,"biography":434,"rating":381,"total_answer":203},"Rasheed","https://api.math-master.org/img/experts/5/5.webp",[],[],"Born with an insatiable curiosity, I began my journey with mathematics during my school years. From solving basic arithmetic problems to unraveling complex algebraic equations, I developed a deep passion for the subject. Upon entering university, my learnings diving into advanced topics like abstract algebra, numerical analysis and differential equations.\nIn retrospect, my journey through the realm of mathematics has been an exhilarating adventure of discovery, learning, and growth.\n",{"id":436,"name":437,"photo":438,"certificates":439,"diplomas":440,"biography":441,"rating":381,"total_answer":332},34,"Dexter","https://api.math-master.org/img/experts/34/34.webp",[],[],"I had done my schooling in SPHS, one of premier academic institutions in the city which covered all courses upto K-12. Mathematics had always been my forte, love & passion which earned me a good amount of reputation and accreditation at various phases of my career. My penchant in Maths was observed by my professors in school who did not hesitate to shower me the honour of one of the best mathematics students taught by them. I had a straight A in Mathematics and related sciences right from school to university level. After crossing the barrier of high school with elan, I did not hesitate to continue my graduation in Mathematics major from JU which enlists within the top 5 universities in the country. I stood first in my faculty in the final year of graduation with 85% score amounting to GPA 4.0. During the 3 year graduation, I silently earned the recognition of the top Mathematics student of the class in the eyes of my professors and my fellow students who used to crowd around me when they got stuck in any topic of maths. Overall my journey as a student of Mathematics was challenging, rewarding, enriching and a superbly enjoyable phase of my career. ",{"id":10,"name":443,"photo":444,"certificates":445,"diplomas":446,"biography":448,"rating":449,"total_answer":179},"Corbin","https://api.math-master.org/img/experts/2/2.webp",[],[447],"https://api.math-master.org/img/experts/2/diplomas/1ea09f59-7559-4ce7-8bad-50bcc43ea9ed.webp","My names is James Agustin Molina, I am from Philippines. Back when I in high school, I was not the best in Math in my class; but, If I were to compare my math skills to my other subjects, I think Math is my best subject. I always find solving and learning math techniques fun. Because I think my Math is the my best subject and I always fascinated in how things work, I took engineering in my college. In college, I really enjoyed math and thinking of pursuing a teaching career. I also took and passed my licensure exam right after graduation. Currently, I am taking my Masters Degree in Mathematics.",4.6,{"id":451,"name":452,"photo":453,"certificates":454,"diplomas":455,"biography":459,"rating":449,"total_answer":206},29,"Eliseo","https://api.math-master.org/img/experts/29/29.webp",[],[456,457,458],"https://api.math-master.org/img/experts/29/diplomas/458 2.webp","https://api.math-master.org/img/experts/29/diplomas/458 d.webp","https://api.math-master.org/img/experts/29/diplomas/458 dd.webp","In my school days I was not good in Mathematics but there was something which always attracted me towards Mathematics as playing with numbers and doing round of questions of different exercises of workbook was fun for me and as a hobby I inclined to Math's. I never thought of teaching mathematics and making a career out of it but somethings bound to happen.",{"id":461,"name":462,"photo":463,"certificates":464,"diplomas":465,"biography":466,"rating":449,"total_answer":467},30,"Bud","https://api.math-master.org/img/experts/30/30.webp",[],[],"I started my mathematics journey when i was in school and i went for a lot of competitions in mathematics back then and after worth, i proceeded into the university to study Electronics and Electrical Engineering and the course of studying it, i came across a lot of mathematics course which grounded me alot in mathematics.",95,{"id":469,"name":470,"photo":471,"certificates":472,"diplomas":473,"biography":476,"rating":449,"total_answer":477},31,"Frederik","https://api.math-master.org/img/experts/31/31.webp",[],[474,475],"https://api.math-master.org/img/experts/31/diplomas/344.webp","https://api.math-master.org/img/experts/31/diplomas/344D.webp","Hi, my name is Nitin,\r\nI started taking an interest in maths, because of my grandfather, who was a maths professor at HNBGU University, initially, I was not that good at maths, but with practice, I improved a lot, and I became pretty good at it when I got excellent marks in class 10th maths, I chose maths as my primary subject I completed my college and graduation, currently I am pursuing my master, I loved teaching maths, I also teach maths in a local school, and I want to become a professor in maths just like my grandfather.",97,{"id":479,"name":480,"photo":481,"certificates":482,"diplomas":484,"biography":486,"rating":449,"total_answer":487},21,"Jon","https://api.math-master.org/img/experts/21/21.webp",[483],"https://api.math-master.org/img/experts/21/certificates/41e2a05c-d94f-4bcc-9258-f2a7fe4c3c02.webp",[485],"https://api.math-master.org/img/experts/21/diplomas/9258cda6-9255-4b34-8806-23628b0a8153.webp","I passed my matric from Govt.City Muslim school Faisalabad and got first division in school by getting 981/1000 marks.\nAfter matric i got admission in Shiblee Colege Faisalabad and got 883/1100 marks in degree of inter with Math.\nNow currently i'm student of BS Math and running my last semester.",107,{"$sicons":489},{"bxl:facebook-circle":490,"bxl:instagram":494,"mdi:web":496,"la:apple":498,"ph:google-logo-bold":501,"ph:google-logo":504},{"left":491,"top":491,"width":492,"height":492,"rotate":491,"vFlip":184,"hFlip":184,"body":493},0,24,"\u003Cpath fill=\"currentColor\" d=\"M12.001 2.002c-5.522 0-9.999 4.477-9.999 9.999c0 4.99 3.656 9.126 8.437 9.879v-6.988h-2.54v-2.891h2.54V9.798c0-2.508 1.493-3.891 3.776-3.891c1.094 0 2.24.195 2.24.195v2.459h-1.264c-1.24 0-1.628.772-1.628 1.563v1.875h2.771l-.443 2.891h-2.328v6.988C18.344 21.129 22 16.992 22 12.001c0-5.522-4.477-9.999-9.999-9.999\"/>",{"left":491,"top":491,"width":492,"height":492,"rotate":491,"vFlip":184,"hFlip":184,"body":495},"\u003Cpath fill=\"currentColor\" d=\"M11.999 7.377a4.623 4.623 0 1 0 0 9.248a4.623 4.623 0 0 0 0-9.248m0 7.627a3.004 3.004 0 1 1 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