Question

Due to a lightning strike, a forest fire begins to burn and is spreading outward in a shape that is roughly circular. The radius of the circle is modeled by the function r(t) = 2t + 1, where t is the time in minutes and r is measured in meters.

1). Write a function for the area burned by the fire directly as a function of t by computing (A ◦ r)(t).

2) Find the area of the circular burn after 20 minutes.

Answer 1

1 -
`A(t)=\pi (2t+1)^2` meter²

2 - 5,281 meter²

Explanation:

We are given that,

Radius of the circle is modeled by the function,
`r(t)=2t+1`, where 't' is the time in minutes.

Part 1: It is required to compute the area of the forest burned.

Since, Area of the circle =
`A=\pi (r)^2`

So,
`(A\circ r)(t)=A(r(t))`

i.e.
`(A\circ r)(t)=A(2t+1)`

i.e.
`(A\circ r)(t)=\pi (2t+1)^2`

Thus, the area of the forest burned by the fire is
`A(t)=\pi (2t+1)^2` meter²

Part 2: It is required to find the area after 20 minutes of burn.

That is, t = 20 mins.

So substituting, we get,

`A(20)=\pi (2* 20+1)^2`

i.e.
`A(20)=\pi (40+1)^2`

i.e.
`A(20)=\pi (41)^2`

i.e.
`A(20)=\pi * 1681`

i.e.
`A(20)=5281` meter²

Thus, the area burned after 20 mins of fire is 5,281 meter².