Question

1. You’ve been hired by a company that makes rectangular storage containers. Each container has a square base and each container must have a volume of 10 m^3. The material for the base and top costs $6/m^2 and the material for the four sides costs $4/m^2. Find the dimensions of the container that minimize the cost of each container.

2. You want to impress your boss by creating a general strategy for finding the dimensions of the cheapest storage container given that the cost of material for the top and bottom is p dollars per square meter and the cost of the material for the sides is q dollars per square meter. As before, the storage container is rectangular, has a square base, and must have a volume of 10 m^3. Your dimensions will involve p and q.

For each part, you want to prove to your boss that these dimensions actually do minimize the cost.

Accordingly, you include an argument as to why these dimensions MUST minimize the cost.

Answer 1

We have to find cost of box in terms of single variable and then minimize it by using derivative

For relative maximum or minimum

To maximize or minimize a function f(x) 1st we need to find its critical points

For critical points we need to put differentiation of f(x) equal to zero i.e. f '(x) = 0

Then check second derivative of f(x) i.e. f ''(x) is positive or negative at critical numbers If f ''(x) >0 then that gives minimum value of f(x) and if f ''(x) <0 then that give maximum value of f(x)