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 + 4 ), express the area (( a )) burned as a function of time, (( t )) (minutes). ( a(t) = )

Explanation:

Step1: Recall the area formula for a circle

The area \( A \) of a circle is given by \( A = \pi r^2 \), where \( r \) is the radius of the circle.

Step2: Substitute the radius function into the area formula

We are given that the radius as a function of time \( t \) is \( r(t) = 4t + 4 \). We substitute \( r = 4t + 4 \) into the area formula. So we have:
\( A(t)=\pi(4t + 4)^2 \)

Step3: Expand the square

We expand \( (4t + 4)^2 \) using the formula \( (a + b)^2=a^{2}+2ab + b^{2} \), where \( a = 4t \) and \( b = 4 \).
\( (4t+4)^{2}=(4t)^{2}+2\times(4t)\times4 + 4^{2}=16t^{2}+32t + 16 \)

Step4: Multiply by \( \pi \)

Now we multiply the expanded form by \( \pi \):
\( A(t)=\pi(16t^{2}+32t + 16)=16\pi t^{2}+32\pi t+16\pi \)

Answer:

\( 16\pi t^{2}+32\pi t + 16\pi \) (or factored form \( \pi(4t + 4)^{2} \))