Question

I think it's 50 now

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

Here barrel of tomato sauce has spilled on a tile floor.

The function
`r(t)=2t` represents how the sauce is flowing, 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 and the area of pattern is expressed as
`A(r)=\pi* r^2`

A. Now we have to find the area of circle of spilled sauce as a function of time:

`A(r(t))`.

Now, we know that

`A=\pi r^2`

and
`r(t)=2t`

plugging the value of 'r' as function of time in the area of the pattern, we get:

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

So the area of the circular pattern as a function of time is given as:

`A=4 \pi t^2`

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

plugging the value of 't' in the equation of the area as a function of time,
`t=5`

we get:

`A(t)=4 * \pi * (5)^2=4 \pi * 25=100 \pi=314`

(we have taken
`\pi=3.14`)

Therefore,
`A=314` square units