Question

A rectangular fence has to be built from both wood and metal so that opposite parallel sides of the fence are made from the same type of material. The wood costs $10 per foot and the metal costs $20 per foot of the fence used. If you only have $400 to spend on material, what dimensions of the fence will enclose the largest area?

Answer 1

if the dimensions for metal and wood are m and w, then we know that

cost = 2(10w+20m)=400
area = m*w
= m(200-20m)/10
= 20m-2m^2

This is just a parabola with its vertex at m=5

So, the maximum area is with dimensions
metal=5 ft
wood=10 ft