Math Question

Question

A play court on the school playground is shaped like a square joined byba semicircle. The perimeter around the entire playcourt is 182.8 ft., and 62.8 ft. of the total perimeter comes from the semicircle.

 

1) What is the radius of the semicircle? Use 3.14 for pi.

 

2) The school wants to cover the play court with sports court flooring. Using 3.14 for pi, how many square feet of flooring does the school need to purchase to cover the play court?

Answer 1

1) To find the radius of the semicircle, we have to use the information already given about the semicircle and somehow use that to tell us more about it. Since we know the perimeter of the semicircle, we will use the perimeter (or circumference) formula for circles, which is pi*r*2=Circumference. But since this is a semicircle, we only want the perimeter for half a circle, so we can instead use simply pi*r, since we are dividing the circumference formula by 2, thus eliminating the "*2". 

 

We know that the circumference is 62.8, so we plug that into the equation, using 3.14 for pi...

 

3.14*r=62.8

 

Solving for r we get...

 

r=20.

 

2) For part two, we do not want to use circumference or perimeter formulas; instead, we will use area formulas because we are looking for how much space, or area, the entire court takes up. There is no formula for this sort of shape, so instead we must find the area of the semicircle and the square and then add them together. First, the semicircle.

 

The area for a circle is defined by this equation: Area=pi*r². We just found the radius in the previous part (which was 20), so we simply plug that number into the formula along with 3.14 for pi...

 

Area=3.14*20²

Area=1256

 

But this is for a whole circle, and we only want half a circle, so we will divide this number by 2 to end with an area of 628 for the semicircle.

 

For the square, the area can be found by using this formula: Area=L*W. Since the length and width, will be the same in square, we can simply use Area=L². We must now use the information given to find the length of one side of the square.

 

Since the perimeter of the whole court is 182.8 feet and 62.8 feet of that belongs to the semicircle, we know that 182.8-62.8=120. So the part of the perimeter that belongs to the square is 120 feet. Normally we might divide this number by 4 to find one side because a square has 4 sides, but since the semicircle is attached to one side of the square, that side is not included in the perimeter. Therefore, only three sides of the square make up that 120 feet. So we divide 120 by 3 yielding 40 feet per side. 

 

Now, we square 40, giving 1600. So the area of the square is 1600. Now we simply add the area of the square to the area of the semicircle to find the total area for the playcourt...

 

1600+628=2228.

 

So the school would need to buy 2228 square feet of flooring to cover the play court.