Question

HELP PLEASE ASAP!!!

A bucket of paint has spilled on a tile floor. The paint flow can be expressed with the function p(t) = 5t, where t represents time in minutes and p represents how far the paint is spreading.

The flowing paint is creating a circular pattern on the tile. The area of the pattern can be expressed as A(p) = πp2.

Part A: Find the area of the circle of spilled paint as a function of time, or A[p(t)]. Show your work. (6 points)

Part B: How large is the area of spilled paint after 2 minutes? You may use 3.14 to approximate π in this problem. (4 points)

Answer 1

A. It's a composite function, so basically, wherever you see a p, replace it with 5t, because we are given that information. So, your answer is:

`A[p(t)] = 5t \pi 2=10t \pi`

B. Let's use the function we created, and just plug in 2 for t:

`A[p(2)] = 10(2) \pi`

`A[p(2)] = 62.83`

So, your answer is (approximately) 62.83 units².

Answer 2

The area of the circle of spilled paint as a function of time is
`A(p(t)) = 25 \pi t^2`. The area of spilled paint after 2 minutes is 314.

Explanation:

Consider the provided statement.

The paint flow can be expressed with the function
`p(t) = 5t`.

Where t represents time in minutes and p represents how far the paint is spreading.

The flowing paint is creating a circular pattern on the tile. The area of the pattern can be expressed as:
`A(p) = \pi p^2`.

Part A: Find the area of the circle of spilled paint as a function of time, or A[p(t)].

Substitute
`p = 5t` in
`A(p) = \pi p^2`.

`A(p(t)) = \pi (5t)^2`

`A(p(t)) = 25 \pi t^2`

Hence, the area of the circle of spilled paint as a function of time is
`A(p(t)) = 25 \pi t^2`.

Part B: How large is the area of spilled paint after 2 minutes?

Substitute t = 2 in
`A(p(t)) = 25 \pi t^2`.

`A(2) = 25 \pi (2)^2`

`A(2) = 100 \pi`

Use π = 3.14 in above equation.

`A(2) = 100 * 3.14`

`A(2) = 314`

Hence, the area of spilled paint after 2 minutes is 314.