Final answer:
To find the radius of the semicircle, subtract the length of the square's perimeter from the total perimeter, which leaves the length of the semicircle's perimeter. Use the formula for the perimeter of a semicircle to solve for the radius. To calculate the area needed for sports court flooring, add the areas of the square and semicircle parts of the play court.
Step-by-step explanation:
To solve for the radius of the semicircle, first, we calculate the perimeter of the square portion of the playing court. Since the total perimeter is 182.8 ft and 62.8 ft comes from the semicircle, the perimeter of the square is 182.8 ft - 62.8 ft, which equals 120 ft. Since the playing court is a square joined by a semicircle, we know the square has four equal sides. Thus, one side of the square will be one-fourth of 120 ft, which is 30 ft. To find the radius of the semicircle, we use the fact that the perimeter of a circle is given by 2πr, which means the perimeter of a semicircle will be πr + 2r (half the circumference plus the diameter). Setting this formula equal to 62.8 ft and using 3.14 for π, we solve for the radius r:
πr + 2r = 62.8 ft
(3.14r + 2r) = 62.8 ft
5.14r = 62.8 ft
r ≈ 12.22 ft
Next, we calculate the area needed for the play court. The area of the square is 30 ft × 30 ft = 900 square feet. The area of the semicircle is half the area of a full circle with the same radius, which is given by the formula A = πr²/2. So, the area of the semicircle is (3.14 × 12.22 ft × 12.22 ft)/2 ≈ 234.31 square feet. Therefore, the total area needed for the play court is 900 square feet + 234.31 square feet = 1134.31 square feet.