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

Answer:

1 - $A(t)=\pi (2t+1)^2$ meter²

2 - 5,281 meter²

Step-by-step 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,

$A(20)=\pi (2\times 20+1)^2$

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

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

i.e. $A(20)=\pi \times 1681$

i.e. $A(20)=5281$ meter²

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