To find the area of the top surface of the wedge Devin bought, we need to follow a series of steps.
1. Calculate the Area of the Entire Circle:
- The radius of the circle is given as 5 inches.
- The formula for the area of a circle is [tex]\(A = \pi r^2\)[/tex].
- Plugging in the radius, we have:
[tex]\[
A = \pi (5)^2 = 25\pi
\][/tex]
- Using the approximation [tex]\(\pi \approx 3.14159\)[/tex], the area of the full circle is:
[tex]\[
A \approx 25 \times 3.14159 \approx 78.54 \text{ square inches}
\][/tex]
2. Determine the Fraction of the Circle Represented by the Wedge:
- The central angle of the wedge is given as 60 degrees.
- Since a full circle is 360 degrees, the fraction of the circle that the wedge represents is:
[tex]\[
\text{Fraction} = \frac{60}{360} = \frac{1}{6}
\][/tex]
3. Calculate the Area of the Wedge:
- The area of the wedge is a fraction of the area of the full circle.
- Using the calculated fraction, the area of the wedge is:
[tex]\[
\text{Wedge Area} = \left(\frac{1}{6}\right) \times 78.54 \approx 13.09 \text{ square inches}
\][/tex]
Therefore, the area of the top surface of the wedge Devin bought is approximately [tex]\(13.09\)[/tex] square inches.
