Question

A stone is dropped in a lake, creating a circular ripple that travels outward at a speed of 60 cm/s.

(a) Find a function g that models the radius as a function of time.

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

(c) Find $f\circ g$. What does this function represent?

Short answer

(a) The function g that models the radius as a function of time is $g\left(t\right)=60t$.

(b) The function f that models the area of circle as a function of radius is $f\left(r\right)=\pi r^2$.

(c) The value of$f\circ g=3600\pi t^2$ and this function represents the area of the circle as a function of time.

Step-by-step solution

Part a. Step 1. Interpret the problem.

A stone is dropped in a lake, creating a circular ripple that travels outward at a speed of 60 cm/s.

This means after 1 second the radius of circle is 60 cm.

And after 2 seconds, the radius of circle is 2 times 60 cm.

And after 3 seconds, the radius of circle is 3 times 60 cm.

Part a. Step 2. Define the variables.

Consider the function g that models the radius of the circle after the time tseconds.

Then after 2 seconds, radius of circle is 120 cm and 180 cm after 3 seconds.

Part a. Step 3. Construct the function.

Observe the pattern that after 1 second the radius of circle is 60 cm.

Then after 2 seconds, radius of circle is 120 cm and 180 cm after 3 seconds.

Therefore, the functions$g\left(t\right)$ that represents the radius as a function of time t is expressed below.

$g\left(t\right)=60t$

Thus, the function g that models the radius as a function of time is $g\left(t\right)=60t$.

Part b. Step 1. Interpret the problem.

A stone is dropped in a lake, creating a circular ripple that travels outward at a speed of 60 cm/s.

This means after 1 second the radius of circle is 60 cm.

And after 2 seconds, the radius of circle is 2 times 60 cm.

And after 3 seconds, the radius of circle is 3 times 60 cm.

Part b. Step 2. Define the variables.

Consider the function f that models the area of the circle.

Let radius of the circle is r units.

Part b. Step 3. Construct the function.

Recall that area of circle of radius r is computed by the formula $A=\pi r^2$.

Now, define a function f as a function of r that models the area of the circle $f\left(r\right)$.

Replace A by $f\left(r\right)$.

$f\left(r\right)=\pi r^2$

Thus, the function f that models the area of circle as a function of radius is $f\left(r\right)=\pi r^2$.

Part c. Step 1. Interpret the problem.

A stone is dropped in a lake, creating a circular ripple that travels outward at a speed of 60 cm/s.

This means after 1 second the radius of circle is 60 cm.

And after 2 seconds, the radius of circle is 2 times 60 cm.

And after 3 seconds, the radius of circle is 3 times 60 cm.

Part c. Step 2. Define the functions.

For the value of composite function $f\circ g$, define the functions.

The function f that models the area of circle as a function of radius is $f\left(r\right)=\pi r^2$.

The function g that models the radius as a function of time is $g\left(t\right)=60t$.

Part c. Step 3. Apply the concept of composite function.

The composition of two functions fand g is expressed as $f\circ g$. This is also called composite function. Mathematically it is expressed below.

$\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$

Part c. Step 4. Find and interpret composite function.

Consider the functions$f\left(r\right)=\pi r^2$ and $g\left(t\right)=60t$.

Now, compute the value of $f\circ g$. This function represents the area of circle as a function of time.

$$\begin{gathered} \left(f\circ g\right)\left(t\right)=f\left(g\left(t\right)\right) \\ =f\left(60t\right) \\ =\pi {\left(60t\right)}^2 \\ =3600\pi t^2 \end{gathered}$$

Thus, the value of$f\circ g=3600\pi t^2$ and this function represents the area of the circle as a function of time.