Carol has 4800 ft of fencing to fence in a rectangular horse Corral

Question

How would I find the dimension of the rectangle that maximizes the area of the coral. What is the maximum area?

Answer 1

We use the formula for perimeter and area for rectangle.

 

Let length = x

Let width = y

Perimeter = 4800 ft

 

Using these variables, we can write equations for perimeter and area.

 

2x + 2y = 4800    ----> Perimeter

 

x + y = 2400     ---->  Perimeter

 

Area = xy

 

 

Substituting the perimeter equation into the area formula in terms a single variable, we get

 

Area = x(2400 - x)

 

Area = -x² + 2400x

 

This is the formula of area we will use.  To find the x-coordinate where the maximum area occurs, we use evaluate

 

x = -b / (2a)

 

where:

a =-1

b = 2400

 

 

x = -(2400) / (2 * -1)

x = -2400 / -2

x = 1200

 

This is your length.  To find the width, substitute this value of x into the formula for perimeter.

 

y = 2400 - x

y = 2400 - 1200

y = 1200

 

The width is 1200 ft.

 

Finally, multiply the length and width to get your maximum area.

 

Area = 1200 ft * 1200 ft

Area = 1440000 ft²