Question

A jug of juice is leaking. The juice that leaked spreads in a circular layer. The area of the circular layer of juice is given by A(r)=πr2
. The radius of the circular layer of juice is increasing by 8 millimeters every minute, which is given by the function r(t)=8t
. What function can be used to determine the area of the circular layer of juice after t minutes?

Enter the missing values.

(1 point)
$$

Answer 1

To find the function that determines the area of the circular layer of juice after \( t \) minutes, we start with the formula for the area of a circle:

\[
A(r) = \pi r^2
\]

Given that the radius \( r \) is increasing according to the function:

\[
r(t) = 8t
\]

Now, we can substitute \( r(t) \) into the area function \( A(r) \):

\[
A(t) = A(r(t)) = A(8t) = \pi (8t)^2
\]

Calculating \( (8t)^2 \):

\[
(8t)^2 = 64t^2
\]

Therefore, the area function \( A(t) \) becomes:

\[
A(t) = \pi (64t^2) = 64\pi t^2
\]

So, the function that can be used to determine the area of the circular layer of juice after \( t \) minutes is:

\[
A(t) = 64\pi t^2
\]

Thus, the missing values that you need to fill in are:

- \( 64 \)
- \( \pi \)
- \( t^2 \)

This means the complete expression for the area is \( 64\pi t^2 \).