Optimization

Find two nonnegative numbers \(x\) and \(y\) whose sum is 100 and their product is a maximum.

A company that makes toy firetrucks has a price-demand function given by \(p(x)=-0.02x+45,\) where \(x\) is the number of trucks sold at a price of \(\$p\) each. The company has a weekly cost function, in dollars, given by \(C(x)=12x+1000,\) where \(x\) is the number of trucks manufactured each week. Determine the selling price of each truck that will maximize the company's profit.

Kirby has 1728 square inches of material, and he needs to make a box with a square base and an open top that has the largest volume possible. Find the dimensions and volume of his box.

A manufacturer is going to make rectangular storage containers with open tops. The volume of each container must be 10 cubic meters, and the length of each container will be twice its width. The cost of the material for the four sides of the container is \(\$4\) per square meter. The cost of the material for the bottom of the container is \(\$8\) per square meter. Find the manufacturer's minimum cost of making each container.