Question

A barrel of tomato sauce has spilled on a tile floor. The sauce flow can be expressed with the function r(t) = 2t, where t represents time in minutes and r represents how far the sauce is spreading.

The spilled sauce is creating a circular pattern on the tile. The area of the pattern can be expressed as A(r) = πr2.

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

Part B: How large is the area of spilled sauce after 5 minutes? You may use 3.14 to approximate π in this problem. (4 points)

Answer 1

Part A: A[r(t)] = π(2t)^2 = 4πt^2

Part B: A(5) = 4 * 3.14 * 5^2 = 314

Answer 2

The area of the circle of spilled sauce as a function of time is
`A(r(t)) = 4 \pi t^2`. The area of spilled sauce after 5 minutes is 314.

Explanation:

Consider the provided statement.

The sauce flow can be expressed with the function
`r(t) = 2t`.

Where t represents time in minutes and r represents how far the sauce is spreading.

The spilled sauce is creating a circular pattern and the area of the pattern can be expressed as:
`A(r) = \pi r^2`.

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

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

`A(r(t)) = \pi (2t)^2`

`A(r(t)) = 4 \pi t^2`

Hence, the area of the circle of spilled sauce as a function of time is
`A(r(t)) = 4 \pi t^2`.

Part B: How large is the area of spilled sauce after 5 minutes?

Substitute t = 5 in
`A(r(t)) = 4 \pi t^2`.

`A(5) = 4 \pi (5)^2`

`A(5) = 100 \pi`

Use π = 3.14 in above equation.

`A(5) = 100 * 3.14`

`A(5) = 314`

Hence, the area of spilled sauce after 5 minutes is 314.