Minimizing Perimeter

Question

A winter recreational rental company is fencing in a new storage area. They have two options. They can set it up at the back of the property and fence it in on four sides. Or, they can attach it to the back of their building and fence it in on three sides. The rental company has decided that the storage area needs to be 100 m2 if it is in the back of their property or 98 m2 if it is attached to the back of the building.

Determine the optimal design for each situation

Answer 1

I assume that optimal design means minimum perimeter.

 

For the larger plot: LW = 100 and P = 2(L+W) = 2[L + (100/L)]

dP/dL = 1 - 100/L² = 0 => L = 10 and the minimum perimeter is 40 m

 

For the smaller plot: LW = 98 and P = 2L + W = 2L + (98/L)

dP/dL = 2 - 98/L²  = 0 => L = 7 and  W = 14 and the minimum perimeter is 28 m