A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula

*r*(*t*) = 4*t* + 5,

express the area *A* burned as a function of time, *t* (minutes).

*A*(*t*) =

The area \( A(t) \) of a circle is given by the formula:

\[
A = \pi r^2
\]

In this problem, the radius \( r(t) \) is changing with time, and is given by the equation:

\[
r(t) = 4t + 5
\]

To express the area \( A(t) \) as a function of time, substitute \( r(t) \) into the area formula:

\[
A(t) = \pi (r(t))^2 = \pi (4t + 5)^2
\]

Now, expand the square:

\[
A(t) = \pi (16t^2 + 40t + 25)
\]

So, the area burned as a function of time is:

\[
A(t) = \pi (16t^2 + 40t + 25)
\]

This is the desired expression for the area \( A(t) \) in terms of time \( t \).

Would you like further details or have any questions?

Here are 5 related questions:

1. How does the rate of change of area vary with time?
2. What is the area burned after 10 minutes?
3. How would the equation change if the radius increases at a different rate?
4. Can you find the rate at which the area is increasing at \( t = 2 \) minutes?
5. How would you represent the time when the area reaches a specific value?

**Tip:** To find the rate of change of the area with respect to time, take the derivative of \( A(t) \) with respect to \( t \).

A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula r(t) = 4t + 5, express the area A burned as a function of time, t (minutes).

Explore how to express the area burned in a circular pattern by a forest fire, where the radius expands as r(t) = 4t + 5. Learn step-by-step how to apply the area of a circle formula and quadratic expansion to solve this geometry and calculus problem, suitable for Grades 10-12.

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