Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 11

MATH100 A December 2023

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[hide] Question 11
Let $A$ be the area of a circle with radius $r$. If $r$ changes at a rate of $2$ cm/s, at what rate is the area enclosed by the circle changing when $r=3$ cm?

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!

[show] Hint
Since the radius of the disc is changing with time, so is its area. Thus, $A$ is a composite function. But note that we are not asked to find a formula for $A$ as a function of time, we are only asked to find the rate of change of $A$. Is there a way to relate the rate of change of $A$ with the rate of change of $r$?

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[show] Solution

The area $A$ of a circle as a function of its radius $r$ is given by $A(r)=\pi r^{2}$. We are given that $r$ changes with time, and its rate of change is $r'(t)=2{\text{ cm/s}}$, for all $t$. Thus $A$ is also a function of time, and we can differentiate it with respect to time using the chain rule: a standard abuse of notation is to write this application of the chain rule as $A'(t)=A'(r(t))\cdot r'(t)=2\pi r(t)\cdot r'=4\pi r(t).$ If $t_{0}$ is the point in time such that $r(t_{0})=3{\text{ cm}}$, then $A'(t_{0})=4\pi \cdot 3=12\pi {\text{ cm}}^{2}{\text{/s}}$. Note that, strictly speaking, the function (input-output machine) describing the area in terms of time is not the same as the function describing the area in terms of the radius. Mathematically, once we define $A(r)=\pi r^{2}$, the number $A(t)$ can be nothing other than $\pi t^{2}$. We need a second symbol, like ${\widetilde {A}}(t)$, to denote the composite function $A(r(t))$. This is how the symbol $A'(t)$ needs to be interpreted in the line where we write down the chain rule.