Question

A carton of milk has spilled on a tile floor. The milk flow can be expressed with the function m(t) = 9t, where t represents time in minutes and m represents how far the milk is spreading.

The flowing milk is creating a circular pattern on the tile. The area of the pattern can be expressed as A(m) = πm2.

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

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

Answer 1

part A) means that we have to find a composition of functions A and m
A(m(t))=π(9t)²=9πt²
part B)
A(m(t))=9πt²
A(m(2))=9*3.14*(2)²=113.04

Answer 2

The area of circle of spilled milk as a function of time, or A[m(t)] is
`A[m(t)]=81t^2\pi`. The area of spilled milk after 2 minutes is 1017.36 square unit.

Explanation:

Consider the provided information.

The milk flow can be expressed with the function m(t) = 9t, where t represents time in minutes and m represents how far the milk is spreading.

The area of the pattern can be expressed as A(m) = πm².

Part A:

To find the area of the circle of spilled milk as a function of time, substitute the value of function in area as shown:

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

`A[m(t)]=\pi (9t)^2`

`A[m(t)]=81t^2\pi`

Hence. the area of circle of spilled milk as a function of time, or A[m(t)] is
`A[m(t)]=81t^2\pi`.

Part B:

Substitute t = 2 in the above equation.

`A[m(t)]=81(2)^2\pi`

`A[m(t)]=81* 4* 3.14`

`A[m(t)]=1017.36`

Hence, the area of spilled milk after 2 minutes is 1017.36 square unit.