Question

A container of oil has spilled on a concrete floor. The oil flow can be expressed with the function n(t) = 8t, where t represents time in minutes and n represents how far the oil is spreading. The flowing oil is creating a circular pattern on the concrete. The area of the pattern can be expressed as A(n) = πn2. Part A: Find the area of the circle of spilled oil as a function of time, or A[n(t)]. Show your work. (6 points) Part B: How large is the area of spilled oil after 5 minutes? You may use 3.14 to approximate π in this problem. (4 points)

Answer 1

A) [n(t)] = π(8t)^2

B) π40^2 = 5024

Answer 2

As per the statement:

The oil flow can be expressed with the function

n(t) = 8t ....[1]

where,

t represents the time in minutes and

n represents how far the oil is spreading

The flowing oil is creating a circular pattern on the concrete.

The area of the pattern can be expressed as

`A(n) = \pi n^2.` .....[2]

A.

Area of the circle of spilled oil as a function of time:

`\text{A[n(t)]}= \pi (n(t))^2`

then;

Substitute equation [1] we get;

`\text{A[n(t)]}= \pi (8t)^2`

⇒
`\text{A[n(t)]}= 64 \pi t^2` .....[3]

B.

We have to find how large is the area of spilled oil after 5 minutes.

Substitute t = 5 minutes and
`\pi = 3.14` in [3] we have;

⇒
`\text{A[n(5)]}= 64 \cdot 3.14 \cdot 5^2`

⇒
`\text{A[n(5)]}= 200.96 \cdot 25`

Simplify:

`\text{A[n(5)]}= 5,024`

Therefore, 5,024 large is the area of spilled oil after 5 minutes.