To determine the measure of the angle of a sector given the area and the radius, we can start by using the formula for the area of a sector of a circle:
[tex]\[ \text{Area of Sector} = \frac{\theta}{360} \times \pi \times r^2 \][/tex]
where:
- [tex]\(\theta\)[/tex] is the angle of the sector in degrees,
- [tex]\(r\)[/tex] is the radius of the circle,
- [tex]\(\pi\)[/tex] is a mathematical constant approximately equal to 3.14159.
Given:
- The area of the sector, [tex]\( \text{Area} = 52 \text{ square inches} \)[/tex],
- The radius, [tex]\( r = 10 \text{ inches} \)[/tex],
we need to find the angle [tex]\(\theta\)[/tex].
1. Starting with the formula:
[tex]\[ 52 = \frac{\theta}{360} \times \pi \times 10^2 \][/tex]
2. Simplify the right-hand side:
[tex]\[ 52 = \frac{\theta}{360} \times \pi \times 100 \][/tex]
3. Isolate [tex]\(\theta\)[/tex]:
[tex]\[ 52 = \frac{\theta}{360} \times 100\pi \][/tex]
[tex]\[ 52 = \frac{100\pi\theta}{360} \][/tex]
[tex]\[ 52 = \frac{100\pi\theta}{360} \][/tex]
4. Rearrange the equation to solve for [tex]\(\theta\)[/tex]:
[tex]\[ \theta = \frac{52 \times 360}{100\pi} \][/tex]
5. Perform the multiplication and division:
[tex]\[ \theta = \frac{18720}{100\pi} \][/tex]
6. Recognize that [tex]\(\pi\)[/tex] is approximately 3.14159:
[tex]\[ \theta = \frac{18720}{314.159} \][/tex]
7. Compute the division:
[tex]\[ \theta \approx 59.59 \text{ degrees} \][/tex]
Hence, the measure of the angle of the sector is approximately [tex]\(59.59\)[/tex] degrees. This does not correspond exactly to any of the provided choices, but it is closest to [tex]\(60^\circ\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{60^\circ} \][/tex]
