Easy Calculus

Question

A farmer builds a rectangular grid of pens with 1 row and 8 columns using 200 feet of fencing. What dimensions will maximize the total area of the pen?
Width? Heigth? Maximum area of how square feet?

Answer 1

A farmer builds a rectangular grid of pens with 1 row and 8 columns .

So the total perimeter of the all pens is = 2W + 9H

Given that the total perimeter 200 feet .

2W + 9H = 200 

9H = 200 - 2W 

H = ( 200 - 2W ) / 9

The area of the rectangular grid is A = H*W 

A = W* ( 200 - 2W ) / 9

A = ( 200 W - 2 W² ) / 9 

To find maximum dimensions , equate first derivative to zero .

A ' = ( 200 - 4 W ) / 9 

( 200 - 4 W ) / 9 = 0

200 - 4 W = 0

4 W = 200 

W = 200 / 4

W = 50 feet

So the maximum width is 50 feet .

The maximum height is ( 200 - 2(50)) / 9

                                          = (200 -100) / 9

                                         = 100 / 9

                                         = 11.11 feet 

So the maximum height is 11.11 feet  .