To find the area of a sector given the arc length and the radius, we can use the relationship between the arc length, the radius, and the central angle (in radians). Here's the step-by-step process:
1. Identify the given values:
- Radius ([tex]\( r \)[/tex]) = 15 inches
- Arc length ([tex]\( L \)[/tex]) = 60 inches
2. Calculate the central angle in radians:
The formula for the arc length ([tex]\( L \)[/tex]) is:
[tex]\[
L = r \times \theta
\][/tex]
where [tex]\( \theta \)[/tex] is the central angle in radians.
Rearrange the formula to solve for [tex]\( \theta \)[/tex]:
[tex]\[
\theta = \frac{L}{r}
\][/tex]
Substitute the given values:
[tex]\[
\theta = \frac{60}{15} = 4 \text{ radians}
\][/tex]
3. Calculate the area of the sector:
The formula for the area of a sector ([tex]\( A \)[/tex]) is:
[tex]\[
A = \frac{1}{2} \times r^2 \times \theta
\][/tex]
Substitute the given values and the value of [tex]\( \theta \)[/tex]:
[tex]\[
A = \frac{1}{2} \times 15^2 \times 4
\][/tex]
Simplify the calculation step-by-step:
[tex]\[
A = \frac{1}{2} \times 225 \times 4
\][/tex]
[tex]\[
A = 112.5 \times 4 = 450 \text{ square inches}
\][/tex]
So, the area of the sector is 450 square inches.
The correct answer is:
C. 450 in. [tex]$^2$[/tex]
