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

A farmer builds a rectangular grid of pens with 1 row and 8 columns using 750 feet of fencing. What dimensions will maximize the total area of the pen? The total width of each row of the pens should be feet, the total height of each column of pens should be feet, which gives the maximum area of square feet.

Solution

1 row + 8 columns = 750 ft

2x + 9y = 750

Area = x*y = (750 -9y)*y/2 = (-9/2)y^2 + 375y

Maximum area occurs at vertex of quadratic equation :

y = -b/2a = -(375/-2*4.5) = 41.67 ft

x = (750 -9*41.67)/2 = 187.49 ft

width of each column = 187.49/6 = 31.25 ft

height of each column = 41.67 ft

Area = 31.25*41.67 = 1302.18 sqfeet