Applied optimization

A soup can in the shape of a right circular cylinder is to be made from two materials. The material for the side of the can costs $0.015 per square inch and the material for the lids costs $$0.027$ per square inch. Suppose that we desire to construct a can that has a volume of 16 cubic inches. What dimensions minimize the cost of the can?

1. Draw a picture of the can and label its dimensions with appropriat…

Consider the region in the $x$-$y$ plane that is bounded by the $x$-axis and the function $f(x) = 25-x^2$. Construct a rectangle whose base lies on the $x$-axis and is centered at the origin, and whose sides extend vertically until they intersect the curve $y = 25-x^2$. Which such rectangle has the maximum possible area? Which such rectangle has the greatest perimeter?

(Challenge: answer the same questions in terms of positive par…

A rectangular box with a square bottom and closed top is to be made from two materials. The material for the sides costs $1.50 per square foot and the material for the top and bottom costs $3.00 per square foot. If you are willing to spend $15 on the box, what is the largest volume it can contain? Justify your answer completely using calculus.

A company is designing propane tanks that are cylindrical with hemispherical ends. Assume that the company wants tanks that will hold 1000 cubic feet of gas, and that the ends are more expensive to make, costing $5 per square foot, while the cylindrical barrel between the ends costs $2 per square foot. Use calculus to determine the minimum cost to construct such a tank.