optimization question

Question

A rectangular field aligned exactly along the compass directions is to have an area of 360m2. If the north-south sides cost $30/m (and must be at least 8m long each) and the east-west sides cost $10/m (and must be at least 15m long each), find the dimensions so that the cost is a minimum.

Answer 1

If the north-south sides have length a then the east-west sides have length 360/a.

Cost of the a sides = $60a and the cost of the other sides = $7200/a.

Total cost C=60a+7200/a. Differentiate:

dC/da=60-7200/a²=0 at max or min.

60a²=7200, a²=120m², a=10.95m (approx).

360/√120=32.86m (approx).

NS sides have a length of 10.95m and EW sides have a length of 32.86m.

Note that this equalises the cost of the NS sides and the EW sides ($657.27).