Question

Find the dimensions of the rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 200 feet of fencing

Answer 1

Corral dimensions:

x = 50 ft

y = 33,33 ft

A(max) = 1666,5 ft²

A(pen) = 833,25 ft²

Explanation:

Let´s call "x" and "y" dimensions of the corral

Then the perimeter is:

p = 2*x + 2*y , and in order to get two pens of equal dimensions we need to split by the use of other length y then

p = 2*x + 3*y = 200

y = (200 - 2*x)/3

Area of the corral

A(c) = x*y

Area as function of x is:

A(x) = x * ( 200 - 2*x)/3

A(x) = (200*x - 2*x²)/3

A(x) = (200/3)*x - (2/3)*x²

Taking derivatives on both sides of the equation:

A´(x) = 200/3 - (4/3)*x

A´(x) = 0 200/3 - (4/3)*x = 0 ⇒ 200 - 4*x = 0

x = 50 ft.

y = ( 200 - 100)/3

y = 100/3

y = 33,33 ft

A(max) = 33,33*50

A(max) = 1666,5 ft²

Area of each pen

A₂ = 833,25 ft²