L29 - 1

Lecture 29: Sections 6.1 and 6.2 Applications of Extrema Part I: Optimizing a given functional model 64 √ ex. A product has demand function p(x) = x and cost function C(x) = 0.8x + 600, 0 ≤ x ≤ 4000. Find the price that should be charged to maximize profit.

L29 - 2

ex. The cost function for a product is C(x) = 1.25x2 + 45x + 500. a) Find the production level which will minimize the average cost C(x).

L29 - 3

b) What is the minimum average cost?

c) Find the marginal cost at the production level where average cost is minimized.

L29 - 4

Marginal Cost and Minimum Average Cost We can verify our observation that the level of production that will result in the smallest average cost occurs when C(x) = C 0(x) assuming that C 00(x) > 0 there.

L29 - 5

ex. Total sales S (in hundreds of dollars) of a product are related to the amount of money x spent on advertising according to the function S(x) = −0.002x3 + 0.6x2 + x + 500 where x is measured in hundreds of dollars and 0 ≤ x ≤ 200. Given that S(x) is increasing on (0, 200), find the point of diminishing returns at which the rate of growth in sales is maximized.

L29 - 6

Optimization, Part II: Determining the quantity to be optimized ex. You are asked to construct a rectangular shipping crate open at the top and with a square base to hold 32 cubic feet. Can you find the dimensions that will minimize the amount of material necessary to make the crate?

L29 - 7

To Solve an Optimization Problem 1. Determine and label any unknown quantities; draw a sketch if possible. 2. Find the Primary Function: the function representing the quantity to be optimized. 3. Find the Constraint: an equation relating the variables in (1). 4. Substitute the Constraint into the Primary Function to rewrite as a function of one variable. 5. Use calculus to find the desired absolute maximimum or minimimum, considering the domain of the Primary Function. Be sure to check that your critical number(s) give the correct extreme value. (a) If the domain is a closed interval, you must compare the values of the endpoints with the critical numbers. (b) If the domain is an open interval, use the First or Second derivative test. Check that you have an absolute maximum or minimum.

L29 - 8

ex. A farmer has $1500 available to build a fence along a straight river to create two identical rectangular pastures. The materials for the side parallel to the river cost $6 per foot and the materials for the three sides perpendicular to the river cost $5 per foot. Find the dimensions for which the total area of the pastures will be as large as possible, assuming that no fence is needed along the river.

L29 - 9

ex. An open box is to be made from a sheet of cardboard 8 feet long and 3 feet wide by cutting away identical squares from each corner and folding up the remaining sides. Find the dimensions that will produce a box of maximum volume.

 

 

L29 - 10

ex. When hot dogs at the stadium concession sell for $4.00 each, the concessions manager has noted that an average of 960 sell per game. When the price is raised to $4.50, anticipated sales drop by an average of 60 hot dogs. a) Assuming a linear demand function, find the number of hot dogs that should be sold to maximize revenue.

L29 - 11

b) If the concessionaire has fixed costs of $1400 per game and variable costs are 40 cents per hot dog, find the price she should charge to maximize profit.

L29 - 12

Additional Examples (as time permits) ex. A swimmer is 100 meters from a straight shore. A lifeguard is 300 meters from the point on the shore closest to the swimmer. If she can swim at 3 m/sec and run at 5 m/sec, what path will get her to the swimmer as fast as possible?

L29 - 13

ex. A right triangle is bounded in the first quadrant by the x-axis and the parabola y = 16 − x2. What dimensions will maximize the area of the triangle?

L29 - 14

ex. A cylindrical tube is to be constructed of heavy cardboard with one plastic end to hold 250π cubic in. What radius will minimize the cost of constructing the tube if plastic costs ten cents per square inch and the cardboard costs five cents per square inch?

L29 - 15

ex. After a drug is taken orally, the amount of drug in the bloodstream (in mg/liter) after t hours is given by P (t) = 120(e−0.2t − e−t) units, 0 ≤ t ≤ 12. 1) When is the amount of drug in the bloodstream at its highest level?

2) When is the level decreasing at the fastest rate?

L29 - 16

Now you try it! Problems are on page 16 and 17. 1. A company is producing a new product. The financial department projects costs to be C(x) = 300 + 40x − 0.02x2 and total revenue to be R(x) = 100x − 0.03x2 . (a) Find the production level that will maximize the profit. What price should they charge to maximize profit? (b) Find the marginal revenue and marginal cost at the production level from (a). (c) Show that in general, if profit is maximized at a production level x0 , then marginal revenue equals marginal cost at x0 . 2. Economists have determined that the gross domestic product (GDP) of a small country between 2005 and 2015 can be reasonably modeled by the function G(t) = −0.2t3 + 2.4t2 + 80 where G(t) is measured in billions of dollars and t = 0 in 2005. (a) Find the year in which the GDP is maximized. What is the maximum GDP? (b) Find the year in which the rate of growth of the GDP is maximized. 3. The management of a local Target store has decided to enclose an 800 square foot area outside the building for the garden display. One side will be formed by an external wall of the store, two sides will be constructed of pineboards costing $6 per foot and the side opposite the store will be constructed of fencing that costs $3 per foot. What dimensions of the enclosure will minimize the cost? Let x be the length of the side with fencing. 4. You are required to construct a closed rectangular box with a surface area of 48 square feet so that the length is twice the width. What dimensions will maximize the volume of the box? What is the maximum volume?

L29 - 17

5. You are designing a rectangular poster with print area 50 square inches. If there is to be a 4 inch margin at the top and bottom of the poster and a two inch margin at each side, what overall dimensions will minimize the amount of posterboard required? Let x and y be the width and height respectively of the printed area, and express the overall dimensions in terms of x and y. 6. A tour company offers half-day cruises to a nearby island. Each tour has fixed costs of $560 and additional costs of $14 per passenger. At the current price of $32, they expect an average of 200 passengers per cruise. Past statistics indicate that if the price is raised by $3, 25 fewer passengers will take the cruise. Assuming the demand function is linear, what price should the company charge to maximize profit? 7. When a person coughs, his or her trachea (windpipe) contracts, allowing air to be expelled at a maximum velocity. It can be shown that during a cough the velocity v of the airflow is given by the function v = f (r) = kr2 (R − r), where r is the trachea’s radius during a cough, R is the trachea’s normal radius (both measured in centimeters), and k is a positive constant. Find the radius r for which the velocity of the airflow is the greatest (that is, find the absolute maximum value of v on the interval [0, R]). Your answer will be in terms of R.