Optimization Problems

Optimization Problems An optimization problem seeks to find the largest (the smallest) value of a quantity (such as maximum revenue or minimum surface area) given certain limits to a problem. An optimization problem can usually be expressed as “find the maximum (or minimum) value of some quantity Q under a certain set of given conditions”.

Steps in Solving Optimization Problems: Understand the Problem Draw a Diagram Introduce Notation identifying the quantity to be maximized (or minimized), Q, and the other variables (label the diagram) Express Q as a function of the other variables. Also express any relations or conditions among the other variables with equations. Rewrite Q as a function of one variable, using given relationships and determine the domain D of Q. Find the absolute extrema (maximum of minimum) of Q on domain D (use the Closed Interval Method, if the domain of Q is an closed interval)

The Least Expensive Cable Better Cable Company must provide service to a customer whose house is located 2 miles from the main highway. The nearest connection box for the cable is located 5 miles down the highway from the customers’ driveway. The installation cost is $14 per mile for any cable that is laid from the house to the highway. (The cable may be laid along the driveway to the house or across the field). The cost is $10 per mile when the cable is laid along the highway. Determine where the cable should be laid so that the installation cost as low as possible.

Look at the picture: 2 miles Connection Box 5 miles

How much will the customer have to pay if the cable is laid 5 miles along the highway and 2 miles along the drive to the house? Show your calculations.

Do you think this cable will be the least expensive possibility? Explain your reasoning.

Calculus in England No summer visit to England is complete without having lunch on the sunny Goodge Street. There, for a mere pound coin, you can purchase the best fish and chips you’ve ever tasted from any one of friendly street vendors. One of the reason that the prices are so reasonable is that they give you no silverware, nor even a plate: they just roll up a piece of paper into a cone, and toss your food in. (They do give you a little packet of vinegar, though). Neither a long, skinny cone nor a wide fat cone would hold enough fish and chips to make anybody happy. The vendors must be trained to roll a cone of a perfect size. Many students will never get a chance to see sunny Goodge Street, but by trying to solve the vendors’ problem of optimizing the volume of a cone, we can feel as if we are there now.

Find the maximum volume of this cone. For modeling purposes, assume that the piece of paper is a circle of radius 5 inches, and that we are cutting a wedge out of it whose central angle is Θ. Θ 5 Find the maximum volume of this cone.

The Waste-Free Box There is a traditional problem that goes like this: We want to make an open-topped box from an 8,5 x 11 inch sheet of paper by cutting congruent squares from the corners and folding up the sides. What is the maximum possible volume of such a box? What most people never think about is the fate of those four squares of paper. They don’t have to be wasted. By taping them together, and putting the reluctant structure on a desk, one can make a handsome pen-and-pencil holder, which will be a box with neither top nor bottom. (It will still hold pencils as long as it rests on the desk).

Picture: x 8,5 11 bottom no bottom

1. What is the maximum possible combined volume of an open-topped box plus a handsome pen-and pencil holder that can be made by cutting four squares from an 8,5 x 11 inch sheet of paper?

2. Describe the open-topped box that results from the maximal case 2. Describe the open-topped box that results from the maximal case. Intuitively, why do we get the result we do?

3. Repeat this problem for a 6 x 10 inch piece of paper.