Find the dimensions of the rectangular corral producing the greatest enclosed area split into 3 pens of the same size given 500 feet of fencing.

Answer 1

Always start a problem like this by drawing a picture.

Let's use A and B for the sides of one pen. If the B sides are in line, then the total length of fencing is 4A + 6B. This gives us our first equation:

4A + 6B = 500

The total area of 3 pens is 3*A*B

Rearrange the 1st equation to get A as a function of B:

4A + 6B = 500

4A = -6B + 500

A = (-3/2)B + 125

substitute this into the expression for the total area:

Area = R = 3 * ( (-3/2)B + 125 ) * B

Expand and simplify:

R = ( (-9/2)B + 375 ) * B

R = (-9/2)B² + 375B

To find a maximum (or minimum), take the 1st derivative and set it = to 0:

R' = -9B + 375 = 0

9B = 375

B = 41.67

Use this equation to find A:

A = (-3/2)B + 125

A = 62.5

Answer 2

Let x = length of each of two dividers and each of two boundary sides and y = length of each side perpendicular to the dividers (draw a diagram)

4x + 2y = 500. So, y = (500 - 4x) / 2 = 250 - 2x

Area = xy = x(250 - 2x)

The graph of the area function is a parabola opening downward with x-intercepts at x = 0 and x = 125.

The maximum occurs halfway between the x-intercepts. So, at x = 62.5.

Maximum area when x = 62.5 ft and y = 125 ft.