MATH MADE EASY - MathMaster

\\cdot 6 = 15600\\pi \\text{ inches per minute} \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> v = 15600\\pi \\text{ inches per minute} \u003C/math-field>\u003C/math-field>",649,130,"a-cylindrical-piece-of-material-12-in-in-diameter-is-rotated-on-a-lathe-at-1300-rev-min-what-is-the-tangential-velocity-at-the-surface-of-the-cylinder-value-4-points",{"id":56,"category":48,"text_question":57,"photo_question":50,"text_answer":58,"step_text_answer":8,"step_photo_answer":8,"views":59,"likes":60,"slug":61},538025,"If you pay

$35 for a drug and the selling price is$ 50 what is the percent gross profit","1. **Determine the selling price and cost price**: \u003Cbr />\n Selling Price

$SP$ = \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 50 \u003C/math-field>\u003C/math-field> \u003Cbr />\n Cost Price

$CP$ = \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 35 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. **Calculate the gross profit**: \u003Cbr />\n Gross Profit = SP - CP = \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 50 - 35 = 15 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n3. **Calculate the percent gross profit**: \u003Cbr />\n Percent Gross Profit = \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\left

$\\frac{15}{35} \\right$ \\times 100 = 42.8571\\% \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n4. **Answer**: \u003Cbr />\n The percent gross profit is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 42.8571\\% \u003C/math-field>\u003C/math-field>",1281,256,"if-you-pay-35-for-a-drug-and-the-selling-price-is-50-what-is-the-percent-gross-profit",{"id":63,"category":48,"text_question":64,"photo_question":50,"text_answer":65,"step_text_answer":8,"step_photo_answer":8,"views":66,"likes":67,"slug":68},538024,"Ln

$x2 -6x+5$ x+5) - Ln

$X-5$ + Ln X= Ln2","1. Use properties of logarithms to combine the logarithms on the left side:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\ln\\left

$\\frac{(x^2 - 6x + 5)(x + 5)x}{x - 5}\\right$ = \\ln 2 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Set the arguments equal as the logarithms on both sides have the same base:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{

$x^2 - 6x + 5$

$x + 5$ x}{x - 5} = 2 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Simplify and solve for x:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>

$x^2 - 6x + 5$

$x + 5$ x = 2

$x - 5$ \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Considering only real solutions, plug potential values for x and solve the equation: \u003Cbr />\n\u003Cbr />\n5. Calculate with

$x = 3$ :\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>

$3^2 - 6 \\cdot 3 + 5$

$3 + 5$ \\cdot 3 = 2

$3 - 5$ \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n6. Verify:\u003Cbr />\n\u003Cbr />\n - Evaluate each side of the equation to prove equality:\u003Cbr />\n\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 3 = 3 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nTherefore, the solution is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 3 \u003C/math-field>\u003C/math-field>.",1157,231,"ln-x2-6x-5-x-5-ln-x-5-ln-x-ln2",{"id":70,"category":48,"text_question":71,"photo_question":50,"text_answer":72,"step_text_answer":8,"step_photo_answer":8,"views":73,"likes":74,"slug":75},537998,"Y=-x-6\nY=-x+6\nState the number of solutions. Explain reason","1. We have two equations: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Y = -x - 6\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>Y = -x + 6\u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cbr />\n2. Set the equations equal to each other to solve for

$x$ :\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-x - 6 = -x + 6\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Simplify the equation:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-6 = 6\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. The above statement is false, indicating the lines are parallel and never intersect. Thus, there are no solutions.\u003Cbr />\n\u003Cbr />\n5. Final answer: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\text{No solutions.}\u003C/math-field>\u003C/math-field>",1474,295,"y-x-6-y-x-6-state-the-number-of-solutions-explain-reason",{"id":77,"category":48,"text_question":78,"photo_question":50,"text_answer":79,"step_text_answer":8,"step_photo_answer":8,"views":80,"likes":81,"slug":82},537986,"7y+6=5x\n4y+4=3x\nSolve by elimination","1. Start with the two equations: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 7y + 6 = 5x \u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 4y + 4 = 3x \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n2. Rearrange them to isolate terms involving \$x\$ and \$y\$ on one side of the equation: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 5x - 7y = 6 \u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 3x - 4y = 4 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n3. Multiply the second equation by 5 and the first equation by 3 to make the coefficients of \$x\$ equal: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 15x - 21y = 18 \u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 15x - 20y = 20 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n4. Subtract the second equation from the first: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> (15x - 21y) - (15x - 20y) = 18 - 20 \u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> -y = -2 \u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> y = 2 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\n5. Substitute \$y = 2\$ back into the second equation to find \$x\$: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 4(2) + 4 = 3x \u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 8 + 4 = 3x \u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 12 = 3x \u003C/math-field>\u003C/math-field> \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 4 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cbr />\nThus, the solution to the equations is: \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> y = 2 \u003C/math-field>\u003C/math-field> \u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 4 \u003C/math-field>\u003C/math-field>",947,189,"7y-6-5x-4y-4-3x-solve-by-elimination",{"id":84,"category":48,"text_question":85,"photo_question":50,"text_answer":86,"step_text_answer":8,"step_photo_answer":8,"views":87,"likes":88,"slug":89},537877,"A student prepares 500 mL of a sodium chloride (NaCl) solution by dissolving 5.0 M VI = l\nNa = 22.990\ngrams of NaCLin water. What is the molarity of the NaCl solution?","1. Calculate the molar mass of NaCl:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> M_{NaCl} = 22.990 + 35.453 = 58.443 \\, \\text{g/mol} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Calculate the number of moles of NaCl:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> n = \\frac{5.0 \\, \\text{g}}{58.443 \\, \\text{g/mol}} = 0.08555 \\, \\text{mol} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Convert volume from mL to L:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> V = \\frac{500 \\, \\text{mL}}{1000} = 0.5 \\, \\text{L} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Calculate the molarity:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> M = \\frac{n}{V} = \\frac{0.08555 \\, \\text{mol}}{0.5 \\, \\text{L}} = 0.1711 \\, \\text{M} \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Answer: Molarity of the NaCl solution is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.171\\,\\text{M}\u003C/math-field>\u003C/math-field>",315,63,"a-student-prepares-500-ml-of-a-sodium-chloride-nacl-solution-by-dissolving-5-0-m-vi-l-na-22-990-grams-of-naclin-water-what-is-the-molarity-of-the-nacl-solution",{"id":91,"category":48,"text_question":92,"photo_question":50,"text_answer":93,"step_text_answer":8,"step_photo_answer":8,"views":94,"likes":95,"slug":96},537869,"Write 0.0000002005 in scientific notation","To write 0.0000002005 in scientific notation, follow these steps:\u003Cbr />\n\u003Cbr />\n1. Move the decimal point in 0.0000002005 to the right until only one non-zero digit remains to the left of the decimal point. This gives 2.005.\u003Cbr />\n2. Count the number of places the decimal point was moved. In this case, it was moved 7 places to the right.\u003Cbr />\n3. Write the number as a product of the number found in step 1 and 10 raised to the negative of the number of places moved in step 2.\u003Cbr />\n\u003Cbr />\nThe answer is:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2.005 \\times 10^{-7}\u003C/math-field>\u003C/math-field>",250,50,"write-0-0000002005-in-scientific-notation",{"id":98,"category":48,"text_question":99,"photo_question":50,"text_answer":100,"step_text_answer":8,"step_photo_answer":8,"views":101,"likes":102,"slug":103},537866,"n – 9) (n – 5)=","Solution:\u003Cbr />\n1. Use the distributive property to expand the expression:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(n - 9)(n - 5) = n(n - 5) - 9(n - 5)\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Distribute \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>n\u003C/math-field>\u003C/math-field> and \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-9\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>= n^2 - 5n - 9n + 45\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Combine like terms:\u003Cbr />\n\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>= n^2 - 14n + 45\u003C/math-field>\u003C/math-field>",1302,260,"n-9-n-5",{"id":105,"category":48,"text_question":106,"photo_question":50,"text_answer":107,"step_text_answer":8,"step_photo_answer":8,"views":108,"likes":109,"slug":110},537811,"During the school year, 144 of the school’s students were able to participate in a community outreach. If this number is 4/5 of the total students, how many students were not able to participate?","1. Set up the equation for the number of participating students:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\frac{4}{5}x = 144 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Solve for the total number of students \$ x \$:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 144 \\times \\frac{5}{4} \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> x = 180 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Calculate the number of students who did not participate:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 180 - 144 = 36 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nFinal Answer: 36 students did not participate.",1245,249,"during-the-school-year-144-of-the-school-s-students-were-able-to-participate-in-a-community-outreach-if-this-number-is-4-5-of-the-total-students-how-many-students-were-not-able-to-participate",{"id":112,"category":48,"text_question":113,"photo_question":50,"text_answer":114,"step_text_answer":8,"step_photo_answer":8,"views":115,"likes":116,"slug":117},537756,"Assume that when Human Resource managers are randomly selected, 53% say job applicants should follow up within two weeks. If 6 human resource managers are randomly selected, find the probability that 4 of them say job applicants should follow up within two weeks.","1. Calculate the binomial coefficient \$ \\binom{6}{4} \$:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> \\binom{6}{4} = \\frac{6!}{4!(6-4)!} = \\frac{6 \\times 5}{2 \\times 1} = 15 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Calculate \$ (0.53)^4 \$:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>(0.53)^4=0.53\\times0.53\\times0.53\\times0.53\\approx0.07890481\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Calculate \$ (1-0.53)^{6-4} = (0.47)^2 \$:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> (0.47)^2 = 0.47 \\times 0.47 = 0.2209 \u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>4. Put these into the formula:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>P(X=4)=15\\times0.07890481\\times0.2209\\approx0.26145\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>So, the probability that exactly 4 out of 6 HR managers say job applicants should follow up within two weeks is approximately:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>0.2615\u003C/math-field>\u003C/math-field>",745,149,"assume-that-when-human-resource-managers-are-randomly-selected-53-say-job-applicants-should-follow-up-within-two-weeks-if-6-human-resource-managers-are-randomly-selected-find-the-probability-that",{"first":6,"last":119,"prev":8,"next":10},9,{"current_page":6,"from":6,"last_page":119,"links":121,"path":146,"per_page":147,"to":147,"total":35},[122,125,128,130,132,134,136,139,142,144],{"url":6,"label":123,"active":124},"1",true,{"url":10,"label":126,"active":127},"2",false,{"url":13,"label":129,"active":127},"3",{"url":16,"label":131,"active":127},"4",{"url":19,"label":133,"active":127},"5",{"url":22,"label":135,"active":127},"6",{"url":137,"label":138,"active":127},7,"7",{"url":140,"label":141,"active":127},8,"8",{"url":119,"label":143,"active":127},"9",{"url":10,"label":145,"active":127},"Next 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