Question

a circles radius is 22 cm when the circle begins shrinking so that its radius length decreases at a constant rate of 3 cm per second.
a. write a formula that expresses the radius of the circle, r (in cm), in terms of the number of seconds, t, since the circle started shrinking.

• try again. note that r is the dependent variable and not the name of the function.

b. write a formula that expresses the area of the circle, a (in cm²), in terms of the circles radius, r (in cm).
c. write a formula that expresses the circles area, a (in cm²), in terms of the number of seconds, t, since the circle started shrinking.
d. define a function f that determines the area of the circle, f(t) (in cm²), given the number of seconds, t, since the circle started shrinking.
e. repeat part (c) but assume that the radius of the circle is shrinking at a constant rate of f cm per second.

Explanation:

Step1: Find radius formula

The initial radius is 22 cm and it decreases by 1 cm per second. So the radius $r$ in terms of time $t$ (in seconds) is $r = 22 - t$.

Step2: Recall area - radius formula

The area formula of a circle is $A=\pi r^{2}$.

Step3: Substitute radius into area formula

Substitute $r = 22 - t$ into $A=\pi r^{2}$, we get $A=\pi(22 - t)^{2}$.

Step4: Define the function

The function $f(t)$ that gives the area of the circle in terms of $t$ is $f(t)=\pi(22 - t)^{2}$.

Step5: Generalize for rate $k$

If the radius shrinks at a rate of $k$ cm per second, the radius $r = 22-kt$, and the area $A=\pi(22 - kt)^{2}$.

Answer:

a. $r = 22 - t$
b. $A=\pi r^{2}$
c. $A=\pi(22 - t)^{2}$
d. $f(t)=\pi(22 - t)^{2}$
e. $A=\pi(22 - kt)^{2}$