Jim plans to use 200 feet of fencing materials to enclose a rectangular dog pen. If he uses his house as the fourth side of the rectangular pen, what is the max

Question

Jim plans to use 200 feet of fencing materials to enclose a rectangular dog pen. If he uses his house as the fourth side of the rectangular pen, what is the maximum area he can enclose?

Answer 1

Assuming that there is no restriction on the length of the side of the house, let x be the length of the two sides perpendicular to the side of the house.  The length of the side parallel to the house will be 200 - 2x.  The area of the pen will be A = x(200 - 2x) = 200x - 2x².  Take the derivative of this function, then set it equal to 0.  dA/dx = 200 - 4x.  200 - 4x = 0 --> x = 50.  The maximum area is (50)(100) = 5000 square feet.