Question

The stadium has a form of a rectangle with two semicircles attached to the short sides. The long side of a rectangle is 2 times longer than the short one.

a) Construct the expression for the area of the stadium in terms of x, where x denotes a shorter side of the rectangle

b) Calculate the area if x=210 feet, where x denotes the shorter side. Take π=22/7

Answer 1

The area of the stadium is the sum of the area of a rectangle and two semicircles, expressed in terms of the shorter side x. For x=210 feet, the stadium's area is calculated to be 122850 square feet using the value of π as 22/7.

Step-by-step explanation:

Area of a Stadium with Rectangle and Semicircles

a) To express the area of the stadium in terms of x, we need to consider the rectangle and the two semicircles separately. Let's denote the shorter side of the rectangle as x and the longer side as 2x, given that it is twice as long. The area of the rectangle is simply the product of its sides, which is x × 2x = 2x². Each semicircle has a radius of x/2 (because the diameter is equal to the short side of the rectangle, which is x). Now, the area of a full circle with radius r is πr². Therefore, the area of each semicircle is (1/2) π (x/2)². Since there are two semicircles, their combined area is π (x/2)². The total area of the stadium is the sum of the rectangle and semicircles area: 2x² + π (x/2)².

b) To calculate the area with x = 210 feet and π = 22/7, we substitute the values in the expression obtained in part (a). The area of the rectangle is 2 × 210² and the combined area of the semicircles is 22/7 × (210/2)². After simplifying and calculating, the total area is:

Rectangle: 2 × 210² = 88200 square feet

Semicircles: 22/7 × (210/2)² = 34650 square feet

Total Area: 88200 + 34650 = 122850 square feet

Answer 2

2x²+1/4πx², and 122,850 sq. ft.

Step-by-step explanation:

We know that the short side of the rectangle is x. The long side is 2 times longer than the short side; this means the long side is 2x. This makes the area of the rectangular portion of the stadium

x(2x) = 2x².

We have two semi-circles at each end. The diameter of each semi-circle is x; this makes the radius 1/2x.

The area of a circle is A = πr²; since these are semi-circles, the area would be given by A = 1/2πr². Using 1/2x in place of r, we have

A = 1/2π(1/2x)² = 1/2π(1/4x²) = 1/8πx²

Using 22/7 for π, we have

A = 1/8(22/7)x² = 22/56x²

Since there are 2 semi-circles, this gives us

2(22/56x²) = 44/56x²

Simplifying this, we have

A = 22/28x² = 11/14x²

This gives us the expression

2x²+11/14x² for our expression for the area.

(Without using 22/7 for pi, we have

2x²+2(1/8πx²) = 2x²+2/8πx² = 2x²+1/4πx²

Using 210 for x and 22/7 for π, we have

A = 2(210²)+1/4(22/7)(210²) = 88200+34650= 122,850