Question

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 + 1, express the area burned as a function of time, (t) (minutes). a(t)=____

Explanation:

Step1: Recall the area formula for a circle

The area formula of a circle is $A=\pi r^{2}$.

Step2: Substitute the given radius - function into the area formula

We are given that $r(t)=4t + 1$. Substitute $r = 4t+1$ into the area formula $A(r)=\pi r^{2}$, so $A(t)=\pi(4t + 1)^{2}$.

Step3: Expand the expression

Using the formula $(a + b)^{2}=a^{2}+2ab + b^{2}$, where $a = 4t$ and $b = 1$, we have $(4t + 1)^{2}=(4t)^{2}+2\times4t\times1+1^{2}=16t^{2}+8t + 1$. Then $A(t)=\pi(16t^{2}+8t + 1)=16\pi t^{2}+8\pi t+\pi$.

Answer:

$A(t)=16\pi t^{2}+8\pi t+\pi$