Math CentralQuandaries & Queries


Question from Wendy, a student:

Hi, I have problems solving this problem. We didn't exactly go over these kind of problems and the book doesn't have an example either. Thank you for your help!

A stone is dropped in a lake, creating a circular ripple that travels outward at a speed of 80 cm/s.
(a) Find a function g that models the radius as a function of time t.

g(t) =

(b) Find a function f that models the area of the circle as a function of the radius r.

f(r) =

(c) Find f compose g.
f compose g =

Hi Wendy,

The radius of the circle increases at a rate of 80 cm/s. What is the radius after 1 second? What is the radius after 2 seconds? What is the radius after 5 seconds? What is the radius after $t$ seconds? This is the function $g(t)$ in part (a).

The function $f(r)$ in part (b) is the expression you know for the area of a circle given its radius $r.$

Now that you have $g(t)$ and $f(r)$ if I give you a time, for example 3.5 seconds, you can use $g(t)$ to find the radius of the circle at time 3.5 seconds and then use this radius and $f(r)$ to find the area of the circle. The point of part (c) is to find an expression that you can use with the time of 3.5 seconds to find the area in one step without explicitly finding the radius.

Try the problem now. Write back if you get stuck, tell me what you have done and I will try to help.


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