Lecture : Applications of Maxima and Minima (Optimization) MTH 124

Transcription

1 The motivation to minimize or maximize a function is strictly determined by the context of the problem. For example we may want to maximize the area of a fenced in area given a fixed amount of fencing material. On the other hand we may want to minimize the cost of producing some good. Such problems are a type of optimization problem. When approaching such problems it may be helpful to make the following considerations. Identify any unknowns, particularly what you re trying to optimize(the objective function) and what variables it depends on. Pay careful attention to units, as they are often useful for sense making. Draw a diagram whenever possible to help you understand the problem better. Does the problem have any constraints? If so, identify them and express them as an equation or inequality. Do you need any formulas that are not given? If so, find them and write the down. Once you have your objective function and any constraints reduce the number of variables(if necessary). If a problem involves several variables solve any constraint equations to express all variables in terms of one particular variable. Example 1 A clothing company manufactures jerseys in runs of up to 500. Its cost, in dollars, for a run of x hockey jerseys is given by C(x) = 2, x + 0.2x 2. How many jerseys should the company produce per run in order to minimize average cost? 79

2 Example 2 A company that produces socks has daily operating costs of $50 per laborer and $30 per robot. The number of pairs of socks the company can manufacture in a day is given by the production formula q = 50n 0.6 r 0.4, where q is the number of pairs of socks that can be manufactured by n laborers, and r robots. Assuming the company wants to produce 1,000 pairs of socks per day at a minimum cost, how many laborers and how many robots should it use? 80

3 Example 3 Leroy is building raised planter boxes for his vegetable garden. For each planter box he will use a 32 foot length of red cedar he milled last year. 1 If each box is to be rectangular, what length and width will maximize the area of each box? 1 Red cedar, being naturally resistant to rot, is an excellent choice for such an application! 81

4 1. According to market research, a company that produces strollers determines it can sell a quantity of q = 300, p 2 strollers per year. What price will bring the greatest annual revenue? 82

5 2. You are building a triangular garden along a stream using two sections of fence that meet to form a right angle. The hypotenuse of the triangle is formed by the stream. The fencing of one border is $5 per foot, the fencing of the other border $1 per foot, and no fencing is required along the river. You have a budget of $100 and would like to have the biggest garden possible. What are the dimensions of your garden? What area does it enclose? 83

6 3. A can company has an order to make cylindrical cans. The only requirement is that the can can is cylindrical and has a capacity of cubic centimeters. What height and radius should the company use to minimize the amount of material they use to make the can? 2 2 A can of coke has the same volume used in this problem, but has a radius of cm and a height of cm. Why do you think this doesn t this agree with what you found? 84

7 4. Shanay wants to build a rectangular enclosure for her pet chihuahua. Three sides of the enclosure will be made up of fencing material and the fourth side will be the wall of her garage. Given that she has 100 feet of fencing material available what is the largest area she can enclose? You may assume the garage wall is as long as you need it to be. 85