A rancher has 288 feet of fencing to enclose two adjacent re

A rancher has 288 feet of fencing to enclose two adjacent rectangular corrals. What dimensions will produce the largest total area?

What is the maximum total area?

Solution:Perimeter =288 feet.

Let length of the rectangle be x and bredth be y.

Since there are two adjacent rectangular corrals.

Thus perimeter = 4x+ 3y

or 288 = 4x+ 3y

or 288 - 4x = 3y

or (288 - 4x)/3 = y

We know that

Area (A) = (2 x) (y)

= (2 x) ((288 - 4x)/3)

= (576x - 8x^2) / 3

Differentiate , to get

A\' = (576 - 16x) / 3

= 192 - 16x/3

For maxima or minima, A\' = 0

or 192 - 16x/3 = 0

or 192(3) = 16x

or x = 36 feet

And y = (288 - 4x)/3 = (288 - 4(36)/3

= (144)/3

= 48 feet.

Thus length of the rectangle is 36 feet and bredth be 48 feet. Ans

Note : Here bredth is greater than length but it is less than 2 times of length.