Question

A large shipping crate is to be constructed in a form of a rectangular box with a square base. It is to have a volume of 441 cubic feet. The material for the base of the crate is steel that costs $6 per square foot, the rest of the crate is constructed out of wood. The wood for the top of the crate is less expensive at $3 per square foot and the sides will be constructed from the wood that costs $3.50 per square foot. Find the dimensions that will minimize the cost of this shipping crate.

a) Write the equation(s) that will enable you to complete the task. Determine the cost function to be analyzed and the independent variable.

b) Find the dimensions that will minimize the cost of the shipping crate. Round to the nearest hundredth, label with units.

c) Use Calculus to prove that the dimensions you found will produce the minimum cost of the shipping crate.

d) What is the lowest cost at which this crate can be built? Round to the nearest cent.

Please show supporting work. Thank you.

Answer 1