Optimization Techniques

    An optimization  procedure is a step-by-step procedure to solve an application problem using calculus; and in particular, using derivatives. Here is a typical optimization procedure:

    (i) Introduce mathematical notation.

    (ii) Express information as equations.

    (iii) Formulate the problem mathematically.

    (iv) Solve the mathematical problem.

    (v) Answer the original problem.

We will illustrate this optimization procedure in the following examples.

Example (Optimizing Area) A woman plans to fence off a rectangular garden whose area is 64 What should be the dimensions of the garden if she wants to minimize the amount of fencing used?

    Solution. Let and be the dimensions of the rectangular plot. The fencing (perimeter) is and the area is with domain We want to minimize so we write as a function of one variable, say

Since

we see that with when and also since

we see that Thus, is a minimum; and the dimensions of the garden should be ft by ft.

Example (Optimizing Volume) Find the dimensions of the right circular cylinder of largest volume that can be inscribed in a sphere of radius

Solution. Let be the radius of the sphere and the radius of the cylinder with height so that Since the volume of the cylinder is given by

Since

when Since and give minima, we see by the Extreme Value Theorem that must be a maximum. The dimensions are and

Example (Optimizing an Angle) The bottom of an 8-ft-high mural painted on a vertical wall is 13 ft above the ground. The lens of a camera fixed to a tripod is 4 ft above the ground. How far from the wall should the camera be placed to photograph the mural with the largest possible angle?

Solution. Let the horizontal distance from the camera to the wall be Let be the angle of elevation from the camera lens to the top of the mural and the angle of elevation from the camera to the bottom of the mural. Also, let Then

Since

optimization techniques _gr_42.gif]

we see that and when Since the endpoint is obviously a minimum we have the maximum when or approximately feet.

Example (Optimizing Distance) A truck is 250 mi due east of a sports car and is traveling west at a constant speed of 60 mi/h. Meanwhile, the sports car is going north at 80 mi/h. When will the truck and the car be closest to each other? What is the minimum distance between them?

Solution. Draw a figure with the car at the origin of a Cartesian coordinate system and the truck at At time (in hours) the truck is at position while the car is at Let be the distance that separates them. Then and so that and We will minimize the square of the distance,

Since the derivative of the distance squared is 0 when Substituting into the equation for produces the shortest distance:

Thus,

and so which is the minimum distance (because there is no maximum distance and the Extreme Value Theorem applies).

Example (Optimizing Time) A jeep is on the desert at a point located 40 km from a point , which lies on a long straight road. The driver can travel at 45 km/h on the desert and 75 km//h on the road. The driver will win a prize if he arrives at the finish line at point , 50 km from , in 84 minutes or less. What route should he travel to minimize the time of travel? Does he win the prize?

Solution. Suppose that the driver heads for a point located km down the road from towards his destination. We want to minimize the time. We will need to remember the formula or in terms of time Since the distance between and is and the distance between and is the total time is given by

Since

we find that is the only critical number of To find the extreme values we evaluate at the endpoints, we find

Therefore, the driver can minimize the total driving time by heading for a point that is from the point and then traveling on the road to point He wins the prize because the minimal route is only 83 minutes.

Marginal analysis is concerned with the way quantities such as price, cost, revenue, and profit vary with small changes in the level of production. The demand function is defined to be the price that consumers will pay for each unit of the commodity when units are brought to market. Then is the total revenue function derived from the sale of the units and is the total profit function where is the total cost function for producing units.

Example (Optimizing Profits) A toy manufacturer produces an inexpensive doll (Dolly) and an expensive doll (Polly) in units of hundred and hundred, respectively. Suppose it is possible to produce the dolls in such a way that

and that the company receives twice as much for selling a Polly doll as for selling a Dolly doll. Find the level of production for both and for which total revenue derived from selling these dolls is maximized. What vital assumption must be made about sales in the model?

Example (Optimizing Revenue) A business manager estimates that when dollars are charged for every unit of a product, the sales will be units. At this level of production, the average cost is modelled by

(a) Find the total revenue and total cost functions, and express the profit as a function of

(b) What price should the manufacturer charge to maximize profit? What is the maximum profit?

Example (Optimizing Costs) Suppose the total cost (in dollars) of manufacturing units of a certain commodity is

(a) At what level of production is the average cost per unit the smallest?

(b) At what level of production is the average cost per unit equal tot he marginal cost?

(c) Graph the average cost and the marginal cost on the same set of axes, for

Cite this as:
Optimization Techniques
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/optimization-techniques.html