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

Answer:

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

Answer 2

Let

x= denotes a shorter side of the rectangle 

y= long side of a rectangle

y=2x

r=D/2=x/2

the area of the stadium A=x*y+ π *(x/2)²

A=x*(2x)+π*(x²/4)- ------->2x²+π*x²/4=x²*(2+π/4)

 

a) the area of the stadium in terms of x  is 

A=x²*(2+π/4)

 

b) if x=210 feet  

π=22/7

 

A=(210)²*(2+π/4)------------------ >(210)²*(2+22/28)------ >(210)²*(78/28)

 

A=122850 feet²