§3.6 Applied optimization

Introduction

More applied optimization problems

Summary

According to U.S. postal regulations, the girth plus the length of a parcel sent by mail may not exceed 108 inches, where by “girth” we mean the perimeter of the smallest end. What is the largest possible volume of a rectangular parcel with a square end that can be sent by mail? What are the dimensions of the package of largest volume?

Let $x$ represent the length of one side of the square end and $y$ the length of the longer side. Label these quantities appropriately on the image shown in the provided figure.

Draw a picture of the can and label its dimensions with appropriate variables.

Use your variables to determine expressions for the volume, surface area, and cost of the can.

Determine the total cost function as a function of a single variable. What is the domain on which you should consider this function?

Find the absolute minimum cost and the dimensions that produce this value.

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 farmer wants to start raising cows, horses, goats, and sheep, and desires to have a rectangular pasture for the animals to graze in. However, no two different kinds of animals can graze together. In order to minimize the amount of fencing she will need, she has decided to enclose a large rectangular area and then divide it into four equally sized pens by adding three segments of fence inside the large rectangle that are parallel to two existing sides. She has decided to purchase 7500 ft of fencing. What is the maximum possible area that each of the four pens will enclose?

Two vertical poles of heights 60 ft and 80 ft stand on level ground, with their bases 100 ft apart. A cable that is stretched from the top of one pole to some point on the ground between the poles, and then to the top of the other pole. What is the minimum possible length of cable required? 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.