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Answer :
Final answer:
The area of the yellow portion of the circle after a pentagon is inscribed and painted red is calculated by finding the total area of the circle, calculating the area of the inscribed pentagon, and subtracting the latter from the former. The area of the yellow portion is approximately 328.2 square inches.
Explanation:
First, we need to find the area of the entire circle, which is done using the formula Area = pi * radius^2. Therefore, the area of the circle is π*(15.23)^2 ≈ 730.5 square inches. Next, we need to find the area of a regular pentagon inscribed in a circle. The formula for the area of a regular pentagon with radius r is Area = 2.5 * r * r * sin(2π/5). So, the area of the pentagon is 2.5*15.23^2*sin(2π/5) ≈ 402.3 square inches. Finally, we subtract the area of the pentagon from the area of the circle to find the area of the yellow portions of the circle. The area of the yellow portion is 730.5 - 402.3 = 328.2 square inches.
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