Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 01

MATH102 December 2013

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Question C 01

A mollusc grows inside a shell and produces new shell material at its opening (indicated in the image below in bold) The volume of the shell can be approximated as that of a cone of base radius r and height h. Given that the height of the cone grows at a constant rate of 0.1 cm/year, at what rate will the volume of the cone be changing when $h=10$ and $r=1$ cm? Leave your answer in terms of $\pi$. Note that: $V_{\text{cone}}={\frac {1}{3}}\pi r^{2}h$

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
What relationship between r and h can we infer by similar triangles?

Hint 2
Since we are only given the rate of change in the cone's height, try expressing the volume of the shell (cone) entirely in terms of h.

Hint 3
Differentiate with respect to time to relate the rate of change in volume to the rate of change in height (note that both are functions of time).

Checking a solution serves two purposes: helping you if, after having used all the hints, 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. The volume of a cone is given by $V={\frac {1}{3}}\pi r^{2}h$ By the physical application of the problem, we can assume that the base radius of the cone changes as the height increases. First, we should express the radius of the cone as a function of its height. By similar triangles, and the fact that r = 1 when h = 10, we find that the following relation holds: ${\frac {r}{h}}={\frac {1}{10}}\quad \iff \quad r={\frac {h}{10}}$ Thus the volume of the shell can be expressed entirely in terms of h: ${\begin{aligned}V&={\frac {1}{3}}\pi \left({\frac {h}{10}}\right)^{2}h\\&={\frac {\pi }{300}}h^{3}\end{aligned}}$ The rate of change in the volume of the cone can be computed by differentiating V with respect to t, giving us ${\frac {dV}{dt}}={\frac {\pi }{100}}h^{2}\cdot {\frac {dh}{dt}}$ Since we are told that the height of the cone changes at a constant rate of 0.1 cm/year, the rate of change of in the cone's volume when h = 10 cm and r = 1 cm is given by ${\begin{aligned}{\frac {dV}{dt}}&={\frac {\pi }{100}}(10{\text{cm}})^{2}\cdot 0.1{\frac {\text{cm}}{\text{year}}}\\&={\color {blue}{\frac {\pi }{10}}\,{\frac {{\text{cm}}^{3}}{\text{year}}}}\end{aligned}}$

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