Science:Math Exam Resources/Courses/MATH100/December 2010/Question 03

MATH100 December 2010

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Question 03

Full-Solution Problems. In questions 2-8, justify your answers and show all your work. If a box is provided, write your final answer there. Simplification of answers is not required unless explicitly requested. Two cylindrical swimming pools are being filled simultaneously with water, at exactly the same rate measured in $m^{3}/min$. The smaller pool has a radius of $5m$ and the height of the water in the smaller pool is increasing at a rate of $0.5m/min$. The larger pool has a radius of $8m$. How fast is the height of the water increasing in the larger pool? Your answer must be a specific numerical value.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Clearly define your variables using diagrams or sentences. Which quantities remain constant and which quantities are changing? What data is given and what are you trying to solve for?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Let $V_{1}$, $r_{1}$, and $h_{1}$ be the volume, radius, and depth of the water in the large pool and let $V_{2}$, $r_{2}$, and $h_{2}$ be the volume, radius, and depth of the water in the small pool. We know that $r_{1}=8$m and $r_{2}=5$m. Since the pools are being filled at the same rate (in $m^{3}/min$), ${\frac {dV_{1}}{dt}}={\frac {dV_{2}}{dt}}.$ We know that ${\frac {dh_{2}}{dt}}=0.5m/min$ and we want to find ${\frac {dh_{1}}{dt}}$. By the volume formula for a cylinder, $V_{1}=\pi \,r_{1}^{2}h_{1}$ and $V_{2}=\pi \,r_{2}^{2}h_{2}$. Note that $r_{1}$ and $r_{2}$ are constants. The variables of the problem are $V_{1},V_{2},h_{1},$ and $h_{2}.$ Since ${\frac {dV_{1}}{dt}}={\frac {dV_{2}}{dt}}$, $\pi \,r_{1}^{2}{\frac {dh_{1}}{dt}}=\pi \,r_{2}^{2}{\frac {dh_{2}}{dt}}$. Hence, the water depth in the the larger pool is increasing at a rate of ${\frac {dh_{1}}{dt}}={\frac {r_{2}^{2}}{r_{1}^{2}}}{\frac {dh_{2}}{dt}}={\frac {(5m)^{2}}{(8m)^{2}}}{\frac {1}{2}}m/min={\frac {25}{128}}m/min$.

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