A rancher wants to create two rectangular pens, as shown in the figure, using an existing fence line as one side. The pens need to have a total area of 972 square feet. What dimensions should be used to minimize the amount of fence used?

A rancher wants to create two rectangular pens, as shown in the attached figure

Let the side parallel to the existing fence line = y
And the sides which is perpendicular to existing fence line = x
The pens need to have a total area of 972 square feet
∴ x y = 972 
∴ y = 972/x

Let the length of needed fence = L
∴ L = y + 3x
substituting with the value of y
∴ [tex]L = \frac{972}{x} +3x[/tex]
differentiating the length with respect to x and equating with zero
∴ [tex] \frac{dL}{dx} = \frac{-972}{x^2} +3 = 0[/tex]
solve for x
∴ x = 18
substituting to find y
∴ y = 54

The dimensions should be used to minimize the amount of fence used
is 18 , 54 feet