The park modeled below consists of a rectangular area and 2 semicircular areas. The area of the rectangular part of the park is 60,000 square yards.
Which of the following is closest to the distance along the perimeter of this park?
628 yd
1,856 yd
914 yd
1,228 yd
To find the distance along the perimeter of the park, we need to find the lengths of each side of the rectangular area and the circumference of the semicircular areas.
Let's start by finding the lengths of the sides of the rectangular area.
We are given that the area of the rectangular part of the park is 60,000 square yards. The formula to find the area of a rectangle is length times width, so we can set up the equation:
length * width = 60,000
We don't have enough information to solve for the length and width individually, so we can make an assumption. Let's assume the length is greater than or equal to the width. This means that the length could be any factor of 60,000 that is greater than or equal to the square root of 60,000.
The prime factorization of 60,000 is 2^5 * 3^1 * 5^4. Taking the square root of 60,000, we get approximately 244.95.
So, the possible lengths that are factors of 60,000 and greater than or equal to 244.95 are 300, 400, 500, 600, 750, 800, 1000, 1200, 1500, 2000, 2400, 3000, 4000, 5000, 6000, 7500, 8000, 10000, 12000, 15000, 20000, 24000, 30000, and 60000.
We can eliminate some of these lengths since the park cannot be a line or a square with zero width.
Now, let's find the circumference of the semicircular areas.
The formula to find the circumference of a circle is 2 * π * r, where r is the radius. Since we are given that the semicircular areas are equal, we can find the radius of one of them.
The area of a semicircle is half the area of a full circle, so the area of one semicircular area is 60000 / 2 = 30000 square yards.
The formula to find the area of a circle is π * r^2, so we can set up the equation:
π * r^2 = 30000
We can solve for r:
r^2 = 30000 / π
r ≈ √(30000 / π) ≈ 97.53
The circumference of one semicircular area is approximately 2 * π * r ≈ 2 * 3.14 * 97.53 ≈ 613.81.
Since there are two semicircular areas, the total circumference of the semicircular areas is approximately 613.81 * 2 ≈ 1227.62.
Now, let's find the distance along the perimeter of the park.
The distance along the perimeter is equal to the sum of the lengths of the sides of the rectangular area and the circumference of the semicircular areas.
Let's assume the length of the rectangular area is 800 yards (one of the possible lengths we found earlier) and the width is 75 yards (60,000 / 800). The perimeter of the rectangular area is 2 * (length + width) = 2 * (800 + 75) = 2 * 875 = 1750 yards.
Adding the circumference of the semicircular areas, the total distance along the perimeter is approximately 1750 + 1227.62 ≈ 2977.62.
Among the answer choices, the closest distance along the perimeter is 1,856 yards.