Question

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.

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

Answer 1

I've seen this problem before, many days ago


A.

p(t)=5t
A[p(t)]=A[5t]=π(5t)²=25πt²
A[p(t)]=25πt²


B.
t=5
A[p(t)]=25πt²
A[p(5)]=25π(5)²
A[p(5)]=25π25²
A[p(t)]=625π
A[p(t)]=625(3.14)=1962.5 square units

Answer 2

Given the paint flow can be expressed with the function as:

`p(t) = 5t`

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

It is also given that: 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`

A

To find the area of the circle of spilled paint as a function of time.

`A[p(t)] = A[5t]`

⇒
`A[5t] = \pi \cdot (5t)^2 = 25 \pi t^2`

therefore, the area of the circle of spilled paint as a function of time
`A[p(t)] = 25 \pi t^2`. ......[1]

B

To find how large is the area of spilled paint after 2 minutes.

Substitute t = 2 minutes and
`\pi = 3.14` in [1] we have;

`A[p(2)] = 25 \cdot 3.14 \cdot 2^2 = 100 \cdot 3.14 = 314` square units.

Therefore, 314 square units large is the area of spilled paint after 2 minutes.