Quadratic Equation

Question

A school is fencing in a rectangular area for a playground. It plans to enclose the playground using fencing on three sides, as shown at the right. http://sketchtoy.com/64751880  The school has budgeted enough money for 75 ft of fencing material and would like to make a playground with an area of 600 ft squared. A.Let w represent the width of the playground. Write an expression in terms of w for the length of the playground. B.Write and solve an equation to find the width w. Round to the nearest tenth of a foot. C.What should the length of the playground be?

Answer 1

We need to find the equations for the length of fence needed for 3 sides of the playground. The perimeter will be:

 

2w + l = 75

 

We also know the area, so:

 

w*l = 600

 

For A, we will use the first equation:

 

l = 75 - 2w

 

We substitute that into the second equation for B:

 

w*(75-2w) = 600

75w-2w² = 600

2w²-75w+600 = 0

 

Use the quadratic equation:

 

w = (-(-75)±sqrt((-75)²-4*2*600))/2*2

w = (75±sqrt(5625-4800))/4

w = (75±sqrt(825))/4

w = (75±28.7)/4

w = 25.9 or 11.6

 

The width is either 25.9 or 11.6. To find the length, plug these numbers in to the third equation and we get:

 

l = 75-2(25.9)   or l = 75-2(11.6)

l = 23.2 or l = 51.8

 

So there are 2 solutions that will give us a 75 foot perimeter and an area of 600 feet².