A rancher wishes to enclose a rectangular partitioned corral

A rancher wishes to enclose a rectangular partitioned corral with 1, 800 feet of fencing. (See the illustration.) What dimensions of the corral would enclose the large possible area? (Enter your answers as a comma-separated list.) Find the maximum area. ft^2

Solution

Answer :

Let a be the length and b be the width of the rectangle.

The perimeter of the rectangle is given by 1800 ft.

Therefore , 2( a + b ) =1800

a + b = 900.

Let A be the are of the rectangle then A = ab =a(900 - a)= - a² + 900a.
For the area to be maximized, the 1st derivative of A must be zero.
That is A\' = - 2x + 900 = 0

x = 450 and then y = 900 - 450 = 450.

Hence, the required dimensions to be maximum area is 450 ft, 450 ft

And the maximum area is 450 x 450 = 202,500 ft²