MATH110 April 2011
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Stalagmites are cone-shaped mineral deposits which rise from the floors of limestone caves. They form over millenia as dripping water deposits calcium carbonate onto the cave floor.
Consider a cone-shaped stalagmite with a length always equal to five times its radius. Suppose the stalagmite's height increases at a rate of 0.13 millimetres per year. Write down two expressions for the rate of change of its volume: (i) with respect to its height h, and (ii) with respect to its radius r.
Hint: the volume V of a cone with radius r and height h is equal to .
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.
What fact about the proportions of the stalagmite can you use to relate the variables r and h? Once you have done this, you can write the volume formula either in terms of h (for part i) or r (for part ii).
Try differentiating the equation relating h and r. What does this tell you about the relationship between and ?
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We are given the volume of a cone.
If we differentiate this expression as it is, we will end up with an expression relating to both variables of r and h. To avoid this, we need to write the volume formula in terms of just r or h, by writing one variable in terms of another. The question states that the length of the stalagmite is always five times its radius, so we can write , or, alternatively, .
(i) Finding an expression in terms of h: Using , we plug it into our volume formula to get
And because we know that mm/year, we finally reach an expression relating the rate of change in volume to the height h
(ii) Finding an expression in terms of r: To find an expression in terms of r we simply use the relationship h = 5r to get
To find an expression for solely in terms of , we need to find . We can do this by differentiating the equation to get
Plugging this into our formula for gives
in terms of r, as desired.
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