Loading [MathJax]/jax/output/HTML-CSS/jax.js
dx :","Solution:\u003Cbr />\n1. Identify the function to integrate: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>fx = 5 - 2x + 3x^2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Integrate each term:\u003Cbr />\n- The integral of a constant \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5\u003C/math-field>\u003C/math-field> with respect to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5x\u003C/math-field>\u003C/math-field>.\u003Cbr />\n- The integral of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-2x\u003C/math-field>\u003C/math-field> is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-\\frac{2}{2}x^2 = -x^2\u003C/math-field>\u003C/math-field>.\u003Cbr />\n- The integral of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x^2\u003C/math-field>\u003C/math-field> is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{3}{3}x^3 = x^3\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n3. Combine the integrals:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int 5−2x+3x2 \\, dx = 5x - x^2 + x^3 + C\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Evaluate the definite integral from \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1\u003C/math-field>\u003C/math-field> to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4\u003C/math-field>\u003C/math-field>:\u003Cbr />\n- Compute the antiderivative at \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4\u003C/math-field>\u003C/math-field>: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>F4 = 54 - 4^2 + 4^3 = 20 - 16 + 64 = 68\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Compute the antiderivative at \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1\u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>F1 = 51 - 1^2 + 1^3 = 5 - 1 + 1 = 5\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Find the difference \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>F4 - F1\u003C/math-field>\u003C/math-field>: \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>68 - 5 = 63\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. The value of the integral is:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int_1^4 5−2x+3x2 \\, dx = 63\u003C/math-field>\u003C/math-field>",1389,278,"evaluate-int-1-4-5-2x-3x-2-dx",{"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},537915,"Find the area between y = 9 - x^2 and y = x + 1 from x = -1 to x = 2","Solution:\u003Cbr />\n\u003Cbr />\n1. Given:\u003Cbr />\n- Curve 1: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = 9 - x^2\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Curve 2: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = x + 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Interval for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = -1\u003C/math-field>\u003C/math-field> to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Find points of intersection:\u003Cbr />\n- Set \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = 9 - x^2\u003C/math-field>\u003C/math-field> equal to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = x + 1\u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>9 - x^2 = x + 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n- Rearrange and simplify:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x^2 + x - 8 = 0\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n- Factor the quadratic equation:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x−2x+4 = 0\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n- Solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 2 \\quad \\text{and} \\quad x = -4\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n- Valid intersection within the given interval is at \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 2\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n3. Calculate area between curves from \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = -1\u003C/math-field>\u003C/math-field> to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 2\u003C/math-field>\u003C/math-field>:\u003Cbr />\n- Integral of top curve minus bottom curve:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int_{-1}^{2} (9−x2 - x+1) \\, dx\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= \\int_{-1}^{2} 8−x2−x \\, dx\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Solve the integral:\u003Cbr />\n- Integrate term-by-term:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= \\left[ 8x - \\frac{x^3}{3} - \\frac{x^2}{2} \\right]_{-1}^{2}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n- Evaluate at upper and lower limits:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\left8(2 - \\frac{2^3}{3} - \\frac{2^2}{2} \\right) - \\left8(−1 - \\frac{−1^3}{3} - \\frac{−1^2}{2} \\right)\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n- Simplify and calculate:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= \\left16−frac83−2right - \\left−8+frac13−frac12right\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= 14 - \\frac{8}{3} + 8 - \\frac{1}{3} + \\frac{1}{2}\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= 22 - \\frac{9}{3} + \\frac{3}{6}\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= 22 - 3 + 0.5\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= 19.5\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. The area between the curves from \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = -1\u003C/math-field>\u003C/math-field> to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 2\u003C/math-field>\u003C/math-field> is:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\boxed{19.5}\u003C/math-field>\u003C/math-field>",1091,218,"find-the-area-between-y-9-x-2-and-y-x-1-from-x-1-to-x-2",{"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},537914,"Integration by parts for \\int \\ln x dx :","Solution:\u003Cbr />\n1. Identify the components of integration by parts:\u003Cbr />\n - Let \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>u = \\ln x\u003C/math-field>\u003C/math-field>, then \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>du = \\frac{1}{x} \\, dx\u003C/math-field>\u003C/math-field>.\u003Cbr />\n - Let \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>dv = dx\u003C/math-field>\u003C/math-field>, then \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>v = x\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n2. Apply the integration by parts formula \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int u \\, dv = uv - \\int v \\, du\u003C/math-field>\u003C/math-field>:\u003Cbr />\n * The formula becomes:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int \\ln x \\, dx = x \\ln x - \\int x \\leftfrac1xright \\, dx\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n3. Simplify the remaining integral:\u003Cbr />\n * \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int x \\leftfrac1xright \\, dx = \\int 1 \\, dx = x + C\u003C/math-field>\u003C/math-field>, where \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>C\u003C/math-field>\u003C/math-field> is the integration constant.\u003Cbr />\n\u003Cbr />\n4. Substitute back to get the final result:\u003Cbr />\n * \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\int \\ln x \\, dx = x \\ln x - x + C\u003C/math-field>\u003C/math-field>.\u003Cbr />",258,52,"integration-by-parts-for-int-ln-x-dx",{"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},537913,"show how to do 349-231 in a base 12 system","Solution:\u003Cbr />\n1. Convert the numbers 349 and 231 from base 12 to decimal:\u003Cbr />\n- 349 base 12: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3times122 + 4times121 + 9times120 = 3times144 + 4times12 + 9times1 = 432 + 48 + 9 = 489\u003C/math-field>\u003C/math-field>\u003Cbr />\n- 231 base 12: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2times122 + 3times121 + 1times120 = 2times144 + 3times12 + 1times1 = 288 + 36 + 1 = 325\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Subtract the converted numbers in decimal:\u003Cbr />\n- Subtraction: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>489 - 325 = 164\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Convert the result back to base 12:\u003Cbr />\n- The highest power of 12 in 164 is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>12^1\u003C/math-field>\u003C/math-field> since\u003Cmath−fieldread−onlydefault−mode=\"inline−math\"class=\"math−expression\">\u003Cmath−fieldread−only>122=144\u003C/math−field>\u003C/math−field>istoolarge.\u003Cbr />\n- Divide 164 by 12: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>164 \\div 12 = 13 \\text{ remainder } 8\u003C/math-field>\u003C/math-field>\u003Cbr />\n- 13 in base 12 is represented as \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1 \\times 12 + 1\u003C/math-field>\u003C/math-field> remainder is 1.\u003Cbr />\n- Therefore, 164 in base 12 is 118.\u003Cbr />\n\u003Cbr />\n4. Therefore, 349 base 12 minus 231 base 12 equals: \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>118\u003C/math-field>\u003C/math-field> base 12.",943,189,"show-how-to-do-349-231-in-a-base-12-system",{"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},537912,"show how to do 14+231 in a base 12 system","Solution:\u003Cbr />\n1. Express the numbers in their base 10 equivalent:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>14_{12} = 1 \\cdot 12^1 + 4 \\cdot 12^0 = 12 + 4 = 16\u003C/math-field>\u003C/math-field>\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>231_{12} = 2 \\cdot 12^2 + 3 \\cdot 12^1 + 1 \\cdot 12^0 = 288 + 36 + 1 = 325\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Add the numbers in base 10:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>16 + 325 = 341\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Convert the sum back to base 12:\u003Cbr />\n- To convert 341 to base 12, divide by 12 and keep track of the remainders.\u003Cbr />\n - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>341 \\div 12 = 28\u003C/math-field>\u003C/math-field> with remainder \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5\u003C/math-field>\u003C/math-field> therightmostdigit\u003Cbr />\n - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>28 \\div 12 = 2\u003C/math-field>\u003C/math-field> with remainder \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4\u003C/math-field>\u003C/math-field>\u003Cbr />\n - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2 \\div 12 = 0\u003C/math-field>\u003C/math-field> with remainder \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Read the remainders from last to first to get \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>245_{12}\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\nThus, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>14_{12} + 231_{12} = 245_{12}\u003C/math-field>\u003C/math-field>.",1182,236,"show-how-to-do-14-231-in-a-base-12-system",{"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},537911,"Task 4.1.5pointsThere are several students in the class and some pairs are friends thefriendshiprelationshipissymmetrical. Every student has at least one friend. The whole class came to MatFyz, where each student chooses exactly one of the lectures: either mathematics or computer science. Can you prove that ineachsuchclass the students can divide themselves into two lectures so that each studentZhad at least one friend who attended a different lecture than the student himself Z.","1. Consider the students and friends as a graph where each student is a vertex, and an undirected edge exists between two vertices if both students are friends. \u003Cbr />\n2. The problem requires proving that this graph is bipartite. \u003Cbr />\n3. A graph is bipartite if and only if it has no odd-length cycles.\u003Cbr />\n4. Start with any vertex and perform BFS or DFS to try to color the graph using two colors, such that no two adjacent vertices share the same color. \u003Cbr />\n5. If successful, you have a bipartite graph, meaning that nodes can be divided into two groups, each corresponding to one lecture.\u003Cbr />\n6. If at any point you find a vertex with the same color during the graph traversal across an edge, there exists an odd-length cycle.\u003Cbr />\n7. The presence of such a cycle would contradict the criteria as friendship is symmetrical, ensuring each cycle detected can be colored alternately, validating bipartiteness.\u003Cbr />\n8. Since each student has at least one friend, and that ensures the graph has no isolated points, every path will connect properly into two sets.\u003Cbr />\n9. Thus, the students can always be divided into two lectures maintaining the condition.\u003Cbr />\n\u003Cbr />\nAnswer: The students can always be divided in a manner where each student has at least one friend attending a different lecture.",1006,201,"task-4-1-5-points-there-are-several-students-in-the-class-and-some-pairs-are-friends-the-friendship-relationship-is-symmetrical-every-student-has-at-least-one-friend-the-whole-class-came-to-matfy",{"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},537910,"There are 120 seats in the cinema. Eighth grades come with the same number of students. If three grades come, there will be 54 free seats left. How many seats will be left if four grades come?\n Can you solve the problem in the form of a proportion or ratio?","Ako u dvoranu dolaze tri razredna odjela i ostaje 54 slobodna mjesta, to znači da dolazi \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x\u003C/math-field>\u003C/math-field> učenika, gdje je \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> broj učenika u jednom razredu.\u003Cbr />\n\u003Cbr />\nBroj zauzetih mjesta s tri razreda:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>120 - 54 = 66\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nDakle:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x = 66 \\Rightarrow x = \\frac{66}{3} = 22\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nBroj učenika u jednom razredu je \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>22\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\nAko dolaze četiri razreda to znači da dolazi \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4 \\times 22 = 88\u003C/math-field>\u003C/math-field> učenika.\u003Cbr />\n\u003Cbr />\nBroj slobodnih mjesta s četiri razreda:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>120 - 88 = 32\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nDakle, ostat će \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>32\u003C/math-field>\u003C/math-field> slobodna mjesta.",777,155,"there-are-120-seats-in-the-cinema-eighth-grades-come-with-the-same-number-of-students-if-three-grades-come-there-will-be-54-free-seats-left-how-many-seats-will-be-left-if-four-grades-come-can-yo",{"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},537909,"There are 120 seats in the cinema. Eighth grades come with the same number of students. If three grades come, there will be 54 free seats left. How many seats will be left if four grades come?","1. Izračunaj broja mjesta zauzetih za 3 razreda: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x\u003C/math-field>\u003C/math-field> \u003Cbr />\n2. Postavi jednadžbu: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>120 - 3x = 54\u003C/math-field>\u003C/math-field> \u003Cbr />\n3. Izračunaj broj učenika u jednom razredu: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x = 66\u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = \\frac{66}{3} = 22\u003C/math-field>\u003C/math-field> \u003Cbr />\n4. Izračunaj broj mjesta zauzetih za 4 razreda: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4x = 4 \\times 22 = 88\u003C/math-field>\u003C/math-field> \u003Cbr />\n5. Izračunaj broj slobodnih mjesta za 4 razreda: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>120 - 88 = 32\u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cbr />\nOdgovor: Broj učenika u jednom razredu je 22, a broj slobodnih mjesta kada dođe 4 razreda je 32.",942,188,"there-are-120-seats-in-the-cinema-eighth-grades-come-with-the-same-number-of-students-if-three-grades-come-there-will-be-54-free-seats-left-how-many-seats-will-be-left-if-four-grades-come",{"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},537908,"Determine the probability that when throwing a die, it will land:\n a) number 5\n b) a number less than 7\n c) number 1 or 4","a) The die is a cube with six faces and equal probability for each face. The number 5 appears on one face. Thus, the probability is:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{1 \\text{ favorable outcome}}{6 \\text{ total outcomes}} = \\frac{1}{6} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nb) All numbers on a standard die 1to6 are less than 7, so all outcomes are favorable:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{6 \\text{ favorable outcomes}}{6 \\text{ total outcomes}} = 1 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nc) The numbers 1 and 4 appear, each on one face of the die:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{2 \\text{ favorable outcomes 1or4}}{6 \\text{ total outcomes}} = \\frac{1}{3} \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nAnswer for a): frac16 \u003Cbr />\nAnswer for b): 1 \u003Cbr />\nAnswer for c): frac13",1299,260,"determine-the-probability-that-when-throwing-a-die-it-will-land-a-number-5-b-a-number-less-than-7-c-number-1-or-4",{"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},537907,"Ivan learned 15 out of 20 questions. What is the probability that the student got a question he did not learn?","- Ivan je naučio 15 pitanja od 20, tako da je broj pitanja koja nije naučio \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 20 - 15 = 5 \u003C/math-field>\u003C/math-field>.\u003Cbr />\n- Ukupan broj pitanja je 20.\u003Cbr />\n- Vjerojatnost da je Ivan dobio pitanje koje nije naučio iznosi \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{5}{20} \u003C/math-field>\u003C/math-field>.\u003Cbr />\n- Jednostavnim deljenjem dobijamo vjerojatnost: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{5}{20} = \\frac{1}{4} \u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\nOdgovor je: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{1}{4} \u003C/math-field>\u003C/math-field>.",518,104,"ivan-learned-15-out-of-20-questions-what-is-the-probability-that-the-student-got-a-question-he-did-not-learn",{"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},537906,"561-t+154=137","1. Start with the equation:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>561-t+154=137\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Distribute the negative sign:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>561-t-154=137\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Simplify the left side by combining like terms:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>407-t=137\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Subtract 407 from both sides:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-t=137-407\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Simplify the right side:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-t=-270\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Multiply both sides by -1:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>t=270\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nAnswer: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>t = 270\u003C/math-field>\u003C/math-field>",1257,251,"561-t-154-137",{"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},537905,"431+B−29=678","First, simplify the equation:\u003Cbr />\n\u003Cbr />\n1. Start with the equation: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>431 + B−29 = 678\u003C/math-field>\u003C/math-field>\u003Cbr />\n2. Distribute and simplify: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>431 + B - 29 = 678\u003C/math-field>\u003C/math-field>\u003Cbr />\n3. Combine like terms: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>B + 402 = 678\u003C/math-field>\u003C/math-field>\u003Cbr />\n4. Isolate \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>B\u003C/math-field>\u003C/math-field> by subtracting 402 from both sides: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>B = 678 - 402\u003C/math-field>\u003C/math-field>\u003Cbr />\n5. Calculate the right side: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>B = 276\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nFinal answer: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>B = 276\u003C/math-field>\u003C/math-field>",1140,228,"431-b-29-678",{"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},537904,"Vicente counted 2,367 seashells at the beach. Then he realized he made a mistake and subtracted 400 seashells. How many seashells did he actually count?","1. Start with the original count: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2367\u003C/math-field>\u003C/math-field>.\u003Cbr />\n2. Subtract the adjustment of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>400\u003C/math-field>\u003C/math-field>: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2367 - 400 = 1967\u003C/math-field>\u003C/math-field>.\u003Cbr />\n3. The actual number of seashells counted: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1967\u003C/math-field>\u003C/math-field>.",447,89,"vicente-counted-2-367-seashells-at-the-beach-then-he-realized-he-made-a-mistake-and-subtracted-400-seashells-how-many-seashells-did-he-actually-count",{"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},537903,"Find the critical value Za/2 needed to construct a confidence interval with the given level \n\nLevel 99.7%","1. Identify that the level of confidence is 99.7%.\u003Cbr />\n2. Calculate the area in each tail: \u003Cbr />\n - The confidence level of 99.7% means there's 0.3% in the tails combined.\u003Cbr />\n - The area in one tail is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{0.3\\%}{2} = 0.15\\% \u003C/math-field>\u003C/math-field>.\u003Cbr />\n3. Determine the z-score for the 99.85th percentile since0.15.\u003Cbr />\n4. Look up this value in the standard normal distribution table or using a calculator, the critical z-score is typically approximately 3.00 forastandardnormaldistribution.\u003Cbr />\n5. Conclude that \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> z_{\\frac{\\alpha}{2}} = 3.00 \u003C/math-field>\u003C/math-field> for a 99.7% confidence interval.",805,161,"find-the-critical-value-za-2-needed-to-construct-a-confidence-interval-with-the-given-level-level-99-7",{"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},537902,"i need 3 addition or subtraction problems with 2 or 3 digit number that equal 5","1. Solve \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>12 - 7\u003C/math-field>\u003C/math-field>:\u003Cbr />\n\u003Cbr />\n - Calculate: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>12 - 7 = 5\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n - Answer: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Solve \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>102 - 97\u003C/math-field>\u003C/math-field>:\u003Cbr />\n\u003Cbr />\n - Calculate: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>102 - 97 = 5\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n - Answer: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Solve \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>55 - 50\u003C/math-field>\u003C/math-field>:\u003Cbr />\n\u003Cbr />\n - Calculate: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>55 - 50 = 5\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n - Answer: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5\u003C/math-field>\u003C/math-field>",783,157,"i-need-3-addition-or-subtraction-problems-with-2-or-3-digit-number-that-equal-5",{"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},537901,"i need 3 rounding problems that equal 9","1. Round 8.5 to the nearest integer:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>8.5 \\approx 9\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n2. Round 8.7 to the nearest integer:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>8.7 \\approx 9\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n3. Round 9.2 to the nearest integer:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>9.2 \\approx 9\u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\nAnswer: 9, 9, 9",1229,246,"i-need-3-rounding-problems-that-equal-9",{"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},537900,"72 forth graders are going on a camping trip. 9 chaperones are also going. If a van can seat 12 people how many vans are needed?","1. Calculate the total number of people: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 72 + 9 = 81 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Divide the total number of people by the number of seats per van:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{81}{12} = 6.75 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Round up to the nearest whole number, because you can't have a fraction of a van: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\lceil 6.75 \\rceil = 7 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nTherefore, 7 vans are needed.",801,160,"72-forth-graders-are-going-on-a-camping-trip-9-chaperones-are-also-going-if-a-van-can-seat-12-people-how-many-vans-are-needed",{"id":163,"category":36,"text_question":164,"photo_question":38,"text_answer":165,"step_text_answer":8,"step_photo_answer":8,"views":166,"likes":167,"slug":168},537899,"81 divided by 12","1. Start with the division: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{81}{12} \u003C/math-field>\u003C/math-field>\u003Cbr />\n2. Simplify the fraction: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{81}{12} = \\frac{27}{4} \u003C/math-field>\u003C/math-field>\u003Cbr />\n3. Convert the fraction to a mixed number: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{27}{4} = 6 \\frac{3}{4} \u003C/math-field>\u003C/math-field>\u003Cbr />\n4. Alternatively, divide to get the decimal: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 81 \\div 12 = 6.75 \u003C/math-field>\u003C/math-field>\u003Cbr />\n5. Answer in decimal: 6.75",635,127,"81-divided-by-12",{"id":170,"category":36,"text_question":171,"photo_question":38,"text_answer":172,"step_text_answer":8,"step_photo_answer":8,"views":173,"likes":174,"slug":175},537898,"Factor the expression using GCF 39-13","Solution:\u003Cbr />\n1. Identify the numbers in the expression: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>39\u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>13\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n2. Determine the greatest common factor GCF of the numbers:\u003Cbr />\n- The factors of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>39\u003C/math-field>\u003C/math-field> are: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1, 3, 13, 39\u003C/math-field>\u003C/math-field>\u003Cbr />\n- The factors of \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>13\u003C/math-field>\u003C/math-field> are: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1, 13\u003C/math-field>\u003C/math-field>\u003Cbr />\n- The common factors are: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1, 13\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Therefore, the greatest common factor is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>13\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\n3. Factor the expression by dividing each term by the GCF:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>39 - 13 = 13 \\times 3 - 13 \\times 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Factor out the GCF \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>13\u003C/math-field>\u003C/math-field>: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>133−1\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Simplify the expression inside the parentheses:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3 - 1 = 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Therefore, the factored expression is: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>13 \\times 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Simplified result: \u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>13 \\times 2 = 26\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nThus, the expression \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>39 - 13\u003C/math-field>\u003C/math-field> factors to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>132\u003C/math-field>\u003C/math-field> or simplifies to \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>26\u003C/math-field>\u003C/math-field>.",373,75,"factor-the-expression-using-gcf-39-13",{"first":6,"last":177,"prev":8,"next":10},180,{"current_page":6,"from":6,"last_page":177,"links":179,"path":213,"per_page":214,"to":214,"total":215},[180,183,186,188,190,192,194,197,200,203,206,209,211],{"url":6,"label":181,"active":182},"1",true,{"url":10,"label":184,"active":185},"2",false,{"url":13,"label":187,"active":185},"3",{"url":16,"label":189,"active":185},"4",{"url":19,"label":191,"active":185},"5",{"url":22,"label":193,"active":185},"6",{"url":195,"label":196,"active":185},7,"7",{"url":198,"label":199,"active":185},8,"8",{"url":201,"label":202,"active":185},9,"9",{"url":204,"label":205,"active":185},10,"10",{"url":207,"label":208,"active":185},179,"179",{"url":177,"label":210,"active":185},"180",{"url":10,"label":212,"active":185},"Next »","https://api.math-master.org/api/question",20,3593,{"data":217},{"questions":218},[219,223,227,231,235,239,243,247,251,255,259,263,267,271,275,279,283,287,291,295],{"id":220,"category":36,"text_question":221,"slug":222},531997,"Students Ana Beatriz and Paula decided to register on a website with exercises to study for upcoming simulations, but to register on this website, they need to choose a password consisting of five characters, three numbers and two letters capitalletters. or lowercase). Letters and numbers can be in any position. They know that the alphabet is made up of twenty-six letters and that an uppercase letter differs from a lowercase letter in a password. What is the total number of possible passwords for registering on this site?","students-ana-beatriz-and-paula-decided-to-register-on-a-website-with-exercises-to-study-for-upcoming-simulations-but-to-register-on-this-website-they-need-to-choose-a-password-consisting-of-five-cha",{"id":224,"category":36,"text_question":225,"slug":226},532035,"I want to divide R2200.00betweenAntônio,BetoandCássia,sothatBetoreceiveshalffromAntônioandCássiareceivesathirdofBeto.Undertheseconditions,howmuchmorewillBetoreceivethanCássia?","i−want−to−divide−r−2200−00−between−antonio−beto−and−cassia−so−that−beto−receives−half−from−antonio−and−cassia−receives−a−third−of−beto−under−these−conditions−how−much−more−will−beto−receive−than","id":228,"category":36,"textquestion":229,"slug":230,533908,"Evaluatelimx→∞tan−1(x)\r\nusingthaty=tan−1(x)\r\nexactlywhenx=tan(y)\r\n.(Hint:Bothtan\r\nandtan−1\r\narecontinuous!)","evaluate−limx−tan−1−x−using−that−y−tan−1−x−exactly−when−x−tan−y−hint−both−tan−and−tan−1−are−continuous","id":232,"category":36,"textquestion":233,"slug":234,533920,"solvethefollowingtrigoequationfor0°\u003C=x\u003C=360°.\r\n\r\nsecx=−2","solve−the−following−trigo−equation−for−0−x−360−sec−x−2","id":236,"category":36,"textquestion":237,"slug":238,533960,"B−(−4)=10","b−4−10","id":240,"category":36,"textquestion":241,"slug":242,534004,"Suppose5025 on an admission ticket. He buys 10 games at X dollars each and two boxes of popcorn at Y dollars each. Write an expression to show the total cost of admission game, tickets and popcorn.","john-he-s-going-to-the-carnival-with-his-friends-he-spends-25-on-an-admission-ticket-he-buys-10-games-at-x-dollars-each-and-two-boxes-of-popcorn-at-y-dollars-each-write-an-expression-to-show-the-t",{"id":272,"category":36,"text_question":273,"slug":274},534299,"Three machines called A, B and C, produce 43%, 26% and 31% of the total production of\n a company, respectively. Furthermore, it has been detected that 8%, 2% and 1.6% of the product\n manufactured by these machines is defective.\n a) What is the probability that a product is not defective?\n b) A product is selected at random and found to be defective, what is the\n probability that it was manufactured on machine B?","three-machines-called-a-b-and-c-produce-43-26-and-31-of-the-total-production-of-a-company-respectively-furthermore-it-has-been-detected-that-8-2-and-1-6-of-the-product-manufactured-by-t",{"id":276,"category":36,"text_question":277,"slug":278},534371,"Determine a general formula orformulas for the solution to the following equation. Then, determine the specific solutions ifany on the interval [0,2π).\ncos30=0","determine-a-general-formula-or-formulas-for-the-solution-to-the-following-equation-then-determine-the-specific-solutions-if-any-on-the-interval-0-2-cos30-0",{"id":280,"category":36,"text_question":281,"slug":282},534383,"Find the number of pounds of nails required for 17850 square feet of drywall if each thousand square feet requires 4.5 pounds of nails.","find-the-number-of-pounds-of-nails-required-for-17850-square-feet-of-drywall-if-each-thousand-square-feet-requires-4-5-pounds-of-nails",{"id":284,"category":36,"text_question":285,"slug":286},534466,"A psychologist is investigating the levels of test anxiety in various university courses. Anxiety is measured on a scale ranging from 0 to 100, where 0 indicates the complete absence of anxiety and 100 represents an extreme level of anxiety.\n From the data obtained, it has been discovered that the psychology score is triple that of nursing, and in turn, the latter has a score 10 points lower than the nutrition major. Furthermore, the score in the veterinary degree is 15 points higher than that of nutrition. Finally, if we add the scores of all the races, we will obtain a total of 173 points.\n Pose the equation that represents the situation described in the previous problem and determine: What is the score that psychology obtained regarding its anxiety level before the exams?","a-psychologist-is-investigating-the-levels-of-test-anxiety-in-various-university-courses-anxiety-is-measured-on-a-scale-ranging-from-0-to-100-where-0-indicates-the-complete-absence-of-anxiety-and-10",{"id":288,"category":36,"text_question":289,"slug":290},534513,"Given a circle 𝑘𝑆;𝑟=4𝑐𝑚 and a line |𝐴𝐵| = 2 𝑐𝑚. Determine and construct the set of all\ncenters of circles that touch circle 𝑘 and have radius 𝑟 = |𝐴𝐵|","given-a-circle-k-s-r-4-cm-and-a-line-ab-2-cm-determine-and-construct-the-set-of-all-centers-of-circles-that-touch-circle-k-and-have-radius-r-ab",{"id":292,"category":36,"text_question":293,"slug":294},534541,"8. Measurement Jillian measured the distance around a small fish pond to be 27 yards. What would be a good estimate of the distance across the pond: 14 yards, 9 yards, or 7 yards? Explain how you decided.","8-measurement-jillian-measured-the-distance-around-a-small-fish-pond-to-be-27-yards-what-would-be-a-good-estimate-of-the-distance-across-the-pond-14-yards-9-yards-or-7-yards-explain-how-you-deci",{"id":296,"category":36,"text_question":297,"slug":298},534675,"23,456 + 3,451","23-456-3-451",{"data":300},{"id":301,"category":36,"slug":302,"text_question":303,"photo_question":8,"text_answer":304,"step_text_answer":8,"step_photo_answer":8,"views":305,"likes":306,"expert":307},536249,"walking-along-hadrian-s-wall-a-group-of-walkers-encountered-a-man-who-claimed-discover-a-magical-well-if-a-bag-of-money-was-put-into-the-well-after-counting-to-five-the-well-doubled-the-money-on","Walking along Hadrian's Wall, a group of walkers encountered a man who claimed\n Discover a magical well. If a bag of money was put into the well, after counting to five, the well\n doubled the money on it. Stunned, the walkers asked him: “How much would it cost us to use the\n Good?\" The man replied: \"Every time you put a bag in the well and count to five,\n give me €148. Do you accept?\" They accepted. The man put his bag in the well, counted until\n five and took it out. The money had doubled. The walkers took €148 from the bag, paid and\n He asked to repeat the operation. But the third time they did it, after paying €148\n According to the man, the wallet no longer had any money in it! How is this possible?","1. Let the initials be \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>M_0\u003C/math-field>\u003C/math-field>\u003Cbr />\n2. After the first doubling, the money becomes \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2M_0\u003C/math-field>\u003C/math-field>\u003Cbr />\n3. After paying the fee, they have \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2M_0 - 148\u003C/math-field>\u003C/math-field> remaining\u003Cbr />\n4. After the second doubling, they get \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>22M0−148 = 4M_0 - 296\u003C/math-field>\u003C/math-field>\u003Cbr />\n5. After paying the fee again, they have \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4M_0 - 296 - 148 = 4M_0 - 444\u003C/math-field>\u003C/math-field>\u003Cbr />\n6. After the third doubling, they get \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>24M0−444 = 8M_0 - 888\u003C/math-field>\u003C/math-field>\u003Cbr />\n7. After paying the fee for the last time, they have \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>8M_0 - 888 - 148 = 8M_0 - 1036 = 0\u003C/math-field>\u003C/math-field>\u003Cbr />\n8. Solving for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>M_0\u003C/math-field>\u003C/math-field>:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>8M_0 - 1036 = 0\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>8M_0 = 1036\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>M_0 = \\frac{1036}{8}\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>M_0 = 129.50\u003C/math-field>\u003C/math-field>",1366,273,{"id":308,"name":309,"photo":310,"biography":311,"created_at":8,"updated_at":8,"rating":312,"total_answer":313},15,"Miles","https://api.math-master.org/img/experts/15/15.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",4.9,100,{"data":315},{"questions":316},[317,321,325,329,333,337,341,345,349,353,357,361,365,369,373,377,381,385,389,393],{"id":318,"category":36,"text_question":319,"slug":320},532061,"A car tire can rotate at a frequency of 3000 revolutions per minute. Given that a typical tire radius is 0.5 m, what is the centripetal acceleration of the tire?","a-car-tire-can-rotate-at-a-frequency-of-3000-revolutions-per-minute-given-that-a-typical-tire-radius-is-0-5-m-what-is-the-centripetal-acceleration-of-the-tire",{"id":322,"category":36,"text_question":323,"slug":324},532078,"Solve: −3−2x+23+12=6−4x+9+9.","solve-3-2x-23-12-6-4x-9-9",{"id":326,"category":36,"text_question":327,"slug":328},532309,"Imagine that you are in an electronics store and you want to calculate the final price of a product after applying a discount. The product you are interested in has an original price of 1000MN,but,fortoday,thestoreoffersa253,275 from her investment account. If the\n investment account yields 4% convertible monthly, how much did you have in your investment account one month before making the first withdrawal?\n Sinceyoustartedmakingwithdrawalsyouhavenotmadeanydeposits.","professor-velez-has-withdrawn-40-monthly-payments-of-3-275-from-her-investment-account-if-the-investment-account-yields-4-convertible-monthly-how-much-did-you-have-in-your-investment-account-one",{"id":362,"category":36,"text_question":363,"slug":364},534391,"X+2x+3=4x+18","x-2-x-3-4x-18",{"id":366,"category":36,"text_question":367,"slug":368},534411,"17. A loan for 104259istakenoutfor10yearswithanannualinterestrateof9.4 signs or commas in the answer.","17-a-loan-for-104259-is-taken-out-for-10-years-with-an-annual-interest-rate-of-9-4-compounded-quarterly-what-quarterly-payment-is-required-to-pay-the-loan-off-in-10-years-enter-to-the-nearest",{"id":370,"category":36,"text_question":371,"slug":372},534436,"Nancy is a waitress at Seventh Heaven Hamburgers. She wants to estimate the average amount each table leaves for a tip. A random sample of 5 groups was taken and the amount they left for a tip indollars is listed below:\n\n11.00\t8.00\t6.00\t3.00\t7.00\n\n\na.)Finda90\n \n \u003C \nμ\n \u003C \n\n\n\n\nb.)Ifthesamplesizewerelarger,witheverythingelseremainingthesame,wouldthemarginofErrorincreaseordecrease?\nDecrease\nIncrease\n\n\nc.)IftheConfidencelevelwere95sicons":412},{"bxl:facebook-circle":413,"bxl:instagram":417,"mdi:web":419,"la:apple":421,"ph:google-logo-bold":424,"ph:google-logo":427},{"left":414,"top":414,"width":415,"height":415,"rotate":414,"vFlip":185,"hFlip":185,"body":416},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":414,"top":414,"width":415,"height":415,"rotate":414,"vFlip":185,"hFlip":185,"body":418},"\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 0-6.008a3.004 3.004 0 0 1 0 6.008\"/>\u003Ccircle cx=\"16.806\" cy=\"7.207\" r=\"1.078\" fill=\"currentColor\"/>\u003Cpath fill=\"currentColor\" d=\"M20.533 6.111A4.6 4.6 0 0 0 17.9 3.479a6.6 6.6 0 0 0-2.186-.42c-.963-.042-1.268-.054-3.71-.054s-2.755 0-3.71.054a6.6 6.6 0 0 0-2.184.42a4.6 4.6 0 0 0-2.633 2.632a6.6 6.6 0 0 0-.419 2.186c-.043.962-.056 1.267-.056 3.71s0 2.753.056 3.71c.015.748.156 1.486.419 2.187a4.6 4.6 0 0 0 2.634 2.632a6.6 6.6 0 0 0 2.185.45c.963.042 1.268.055 3.71.055s2.755 0 3.71-.055a6.6 6.6 0 0 0 2.186-.419a4.6 4.6 0 0 0 2.633-2.633c.263-.7.404-1.438.419-2.186c.043-.962.056-1.267.056-3.71s0-2.753-.056-3.71a6.6 6.6 0 0 0-.421-2.217m-1.218 9.532a5 5 0 0 1-.311 1.688a3 3 0 0 1-1.712 1.711a5 5 0 0 1-1.67.311c-.95.044-1.218.055-3.654.055c-2.438 0-2.687 0-3.655-.055a5 5 0 0 1-1.669-.311a3 3 0 0 1-1.719-1.711a5.1 5.1 0 0 1-.311-1.669c-.043-.95-.053-1.218-.053-3.654s0-2.686.053-3.655a5 5 0 0 1 .311-1.687c.305-.789.93-1.41 1.719-1.712a5 5 0 0 1 1.669-.311c.951-.043 1.218-.055 3.655-.055s2.687 0 3.654.055a5 5 0 0 1 1.67.311a3 3 0 0 1 1.712 1.712a5.1 5.1 0 0 1 .311 1.669c.043.951.054 1.218.054 3.655s0 2.698-.043 3.654z\"/>",{"left":414,"top":414,"width":415,"height":415,"rotate":414,"vFlip":185,"hFlip":185,"body":420},"\u003Cpath fill=\"currentColor\" d=\"M16.36 14c.08-.66.14-1.32.14-2s-.06-1.34-.14-2h3.38c.16.64.26 1.31.26 2s-.1 1.36-.26 2m-5.15 5.56c.6-1.11 1.06-2.31 1.38-3.56h2.95a8.03 8.03 0 0 1-4.33 3.56M14.34 14H9.66c-.1-.66-.16-1.32-.16-2s.06-1.35.16-2h4.68c.09.65.16 1.32.16 2s-.07 1.34-.16 2M12 19.96c-.83-1.2-1.5-2.53-1.91-3.96h3.82c-.41 1.43-1.08 2.76-1.91 3.96M8 8H5.08A7.92 7.92 0 0 1 9.4 4.44C8.8 5.55 8.35 6.75 8 8m-2.92 8H8c.35 1.25.8 2.45 1.4 3.56A8 8 0 0 1 5.08 16m-.82-2C4.1 13.36 4 12.69 4 12s.1-1.36.26-2h3.38c-.08.66-.14 1.32-.14 2s.06 1.34.14 2M12 4.03c.83 1.2 1.5 2.54 1.91 3.97h-3.82c.41-1.43 1.08-2.77 1.91-3.97M18.92 8h-2.95a15.7 15.7 0 0 0-1.38-3.56c1.84.63 3.37 1.9 4.33 3.56M12 2C6.47 2 2 6.5 2 12a10 10 0 0 0 10 10a10 10 0 0 0 10-10A10 10 0 0 0 12 2\"/>",{"left":414,"top":414,"width":422,"height":422,"rotate":414,"vFlip":185,"hFlip":185,"body":423},32,"\u003Cpath fill=\"currentColor\" d=\"M20.844 2c-1.64 0-3.297.852-4.407 2.156v.032c-.789.98-1.644 2.527-1.375 4.312c-.128-.05-.136-.035-.28-.094c-.692-.281-1.548-.594-2.563-.594c-3.98 0-7 3.606-7 8.344c0 3.067 1.031 5.942 2.406 8.094c.688 1.078 1.469 1.965 2.281 2.625S11.57 28 12.531 28s1.68-.324 2.219-.563c.54-.238.957-.437 1.75-.437c.715 0 1.078.195 1.625.438c.547.242 1.293.562 2.281.562c1.07 0 1.98-.523 2.719-1.188s1.36-1.519 1.875-2.343c.516-.824.922-1.633 1.219-2.282c.148-.324.258-.593.343-.812s.13-.281.188-.531l.188-.813l-.75-.343a5.3 5.3 0 0 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