A farmer decides to enclose a rectangular​ garden, using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose w

Question

A farmer decides to enclose a rectangular garden, using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose with 100 ft of fence? What should the dimensions of the garden be to give this area?

Answer 1

Let the length of the fence = x

Let the width of the fence = y

 

Perimeter equation is

 

x + 2y = 100

 

Area = xy

 

If we substitute the perimeter equation so that the area is only in terms of y.

 

Area = (100 - 2y)y

Area = 100y - 2y²

 

Now just find the vertex of the parabola

 

Area = -2y² + 100y

Area = -2(y² - 50y + 625) + 1250

Area = -2(y - 25)² + 1250

 

The maximum area is 1250 square feet.