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?
Accepted Solution
A:
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 β΄ x y = 972Β β΄ y = 972/x
Let the length of needed fence = L β΄ L = y + 3x substituting with the value of y β΄ [tex]L = \frac{972}{x} +3x[/tex] differentiating the length with respect to x and equating with zero β΄ [tex] \frac{dL}{dx} = \frac{-972}{x^2} +3 = 0[/tex] solve for x β΄ x = 18 substituting to find y β΄ y = 54
The dimensions should be used to minimize the amount of fence used is 18 , 54 feet