Question

A rectangular storage container with an open top with an open top is to have a volume of 10 m^3. The length of its base is twice the width. Material for the base costs $15 per m^2. Material for the sides costs $2 per m^2. Find the dimensions of the container which will minimize cost and the minimum cost.

What is Base length=

base width=

height=

minimum cost=

Answer 1

If the width is x, then
v = x(2x)*h = 2x^2 h = 10
so h = 5/x^2
The surface area is
2x^2 + 2(xh + 2xh) = 2x^2 + 6xh = 2x^2 + 30/x
The cost is thus
c(x) = 15*2x^2 + 2*30/x = 30(x^2 + 2/x)
dc/dx = 30*2(x^3-1)/x^2
dc/dx = 0 when x = 1
Now finish it off