Answer:
Area = 8,645.48 m²
Frank incorrectly subtracted the area of the square from the area of the four semicircles instead of adding them.
Step-by-step explanation:
The composite figure is composed of a square with side length 58 m and four congruent semicircles with diameters equal to the side length of the square.
Since the diameter of a circle is twice its radius (r), the radius of the semicircles is half the side length of the square, and so:
[tex]r=\dfrac{58\;\sf m}{2}=29\; \sf m[/tex]
The area of a semicircle is half the area of a circle (πr²), and the area of a square is the square of its side length. Therefore:
[tex]\begin{aligned}\textsf{Area of figure}&=4 \times \textsf{Area of a semicircle}+\textsf{Area of a square}\\\\&=4 \times \dfrac{1}{2}\pi r^2 + s^2\end{aligned}[/tex]
Given that the side length of the square is s = 58 and π = 31.4, substitute these values along with r = 29 into the equation, and solve for area:
[tex]\begin{aligned}\textsf{Area of figure}&=4 \times \dfrac{1}{2}\times 3.14\times (29)^2 + (58)^2\\\\&=4 \times \dfrac{1}{2}\times 3.14\times 841 + 3364\\\\&=2\times 3.14\times 841 + 3364\\\\&=6.28\times 841 + 3364\\\\&=5281.48 + 3364\\\\&=8645.48\; \sf m^2\end{aligned}[/tex]
If Frank says the area is 1,917.48 m², then the error he made is subtracting the area of the square from the area of the four semicircles instead of adding them:
[tex]\textsf{Area of 4 semicircles} - \textsf{Area of square}=5281.48-3364=1917.48[/tex]
