And draw the supply and demand curves the same set axes

46 Chapter 1 Functions, Graphs, and Limits

triangle OAB to obtain the required relationship between m1 and m2.]

The highway department is planning to build a picnic area for motorists along a major highway. It is to be rectangular with an area of 5,000 square yards and is to be fenced off on the three sides not adjacent to the highway. Express the number of yards of fencing required as a function of the length of the unfenced side.

Solution
x It is natural to start by introducing two variables, say x and y, to denote the length of the sides of the picnic area (Figure 1.28), and to express the number of yards F of

creasing at increments (�Tbl) of

100, 50, and 10. Finally, use

x

r

If 12 � x � 24, each of the first 12 units costs $1.22, and so the total cost of these 12 units is 1.22(12) � 14.64 dollars. Each of the remaining x � 12 units costs

C(x) � 14.64 � 120 � 50(x � 24) � 50x � 1,065.36

Combining these three formulas, you get

C(x)

150

100

Here is an example from biology.

EXAMPLE 4.4

growth is greatest.

EXAMPLE 4.5

A manufacturer can produce blank video tape at a cost of $2 per cassette. The cas-settes have been selling for $5 apiece, and at that price, consumers have been buy-ing 4,000 cassettes a month. The manufacturer is planning to raise the price of the cassettes and estimates that for each $1 increase in the price, 400 fewer cassettes will be sold each month.

Solution

(a) Begin by stating the desired relationship in words:

� 4,000 � 400(number of $1 increases)

� 4,000 � 400(x � 5)

P(x) � (number of cassettes sold)(profit per cassette)

� (6,000 � 400x)(x � 2)

� $16,900

6,000

4,000

Shortage Surplus

q = D(p)

Actually, an economist’s graph of these functions would not look quite like the one in Figure 1.35. For technical reasons, when dealing with supply and demand curves, economists usually break with mathematical tradition and use the horizontal axis for the dependent variable q and the vertical axis for the independent variable p.

The point of intersection of the supply and demand curves is called the point of market equilibrium. The p coordinate of this point (the equilibrium price) is the market price at which supply equals demand; that is, the market price at which there will be neither a surplus nor a shortage of the commodity.

Since only positive values of p are meaningful in this practical problem, you can con-clude that the equilibrium price is $20. Since the corresponding supply and demand are equal, use the simpler demand equation to compute this quantity to get

D(20) � 410 � 20 � 390

q

q = S(p)

SALES REVENUE 5. Each unit of a certain commodity costs p � 35x � 15 cents when x units of the commodity are produced. If all x units are sold at this price, express the revenue derived from the sales as a function of x.

FENCING 6. A city recreation department plans to build a rectangular playground 3,600 square meters in area. The playground is to be surrounded by a fence. Express the length of the fencing as a function of the length of one of the sides of the playground, draw the graph, and estimate the dimensions of the playground requiring the least amount of fencing.

VOLUME 9. A closed box with a square base has a surface area of 4,000 square centimeters. Express its volume as a function of the length of its base.

In Problems 10 through 14, you will need to know that a cylinder of radius r and height h has volume V � �r2h. Also, recall that a circle of radius r has area A ��r2and circumference C � 2�r.

(b) If the volume V of the can is a constant, express the surface area S in terms of V and r.

PACKAGING 13. A cylindrical can is to hold 4� cubic inches of frozen orange juice. The cost per square inch of constructing the metal top and bottom is twice the cost per square inch of constructing the cardboard side. Express the cost of constructing the can as a function of its radius if the cost of the side is 0.02 cent per square inch.

THE SPREAD OF AN EPIDEMIC 18. The rate at which an epidemic spreads through a community is jointly propor- tional to the number of people who have caught the disease and the number who have not. Express this rate as a function of the number of people who have caught the disease.

POLITICAL CORRUPTION 19. The rate at which people are implicated in a government scandal is jointly pro- portional to the number of people already implicated and the number of people involved who have not yet been implicated. Express this rate as a function of the number of people who have been implicated.

INCOME TAX 23. The following table represents the 1993 federal income tax rate schedule for sin- gle taxpayers.

(a) Express an individual’s income tax as a function of the taxable income x for

ADMISSION FEES 24. A local natural history museum charges admission to groups according to the fol- lowing policy. Groups of fewer than 50 people are charged a rate of $3.50 per person, while groups of 50 people or more are charged a reduced rate of $3 per person.

(a) Express the amount a group will be charged for admission as a function of its size and draw the graph.

material for the sides costs $1 per square meter. Express the construction cost of the box as a function of the length of its base.

CONSTRUCTION COST 26. An open box with a square base is to be built for $48. The sides of the box will cost $3 per square meter, and the base will cost $4 per square meter. Express the volume of the box as a function of the length of its base.

RETAIL SALES 29. A bookstore can obtain a certain gift book from the publisher at a cost of $3 per book. The bookstore has been offering the book at the price of $15 per copy, and at this price, has been selling 200 copies a month. The bookstore is planning to lower its price to stimulate sales and estimates that for each $1 reduction in the price, 20 more books will be sold each month. Express the bookstore’s monthly profit from the sale of this book as a function of the selling price, draw the graph, and estimate the optimal selling price.

RETAIL SALES 30. A manufacturer has been selling lamps at the price of $30 per lamp, and at this price consumers have been buying 3,000 lamps a month. The manufacturer wishes to raise the price and estimates that for each $1 increase in the price, 1,000 fewer lamps will be sold each month. The manufacturer can produce the lamps at a cost of $18 per lamp. Express the manufacturer’s monthly profit as a function of the price that the lamps are sold, draw the graph, and estimate the optimal selling price.

HARVESTING 34. Farmers can get $3 per bushel for their potatoes on July first, and after that, the price drops by 2 cents per bushel per day. On July first, a farmer has 140 bushels of potatoes in the field and estimates that the crop is increasing at the rate of 1 bushel per day. Express the farmer’s revenue from the sale of the potatoes as a function of the time at which the crop is harvested, draw the graph, and estimate when the farmer should harvest the potatoes to maximize revenue.

SUPPLY AND DEMAND 35. The supply and demand functions for a certain commodity are S(p) � 4p � 200 and D(p) � �3p � 480, respectively. Find the equilibrium price and the corre- sponding number of units supplied and demanded, and draw the supply and demand curves on the same set of axes.

(a) Find the equilibrium price and the corresponding number of units supplied and demanded.

PHYSIOLOGY 40. The pupil of the human eye is roughly circular. If the intensity of light I entering the eye is proportional to the area of the pupil, express I as a function of the radius r of the pupil.

RECYCLING 41. To raise money, a service club has been collecting used bottles that it plans to deliver to a local glass company for recycling. Since the project began 80 days ago, the club has collected 24,000 pounds of glass for which the glass company currently offers 1 cent per pound. However, since bottles are accumulating faster than they can be recycled, the company plans to reduce by 1 cent each day the price it will pay for 100 pounds of used glass. Assuming that the club can con- tinue to collect bottles at the same rate and that transportation costs make more than one trip to the glass company unfeasible, express the club’s revenue from its recycling project as a function of the number of additional days the project runs. Draw the graph and estimate when the club should conclude the project and deliver the bottles to maximize its revenue.

where Km is a constant (called the Michaelis constant), Rm is the maximum pos-sible rate, and [S] is the substrate concentration.* Rewrite this equation so that

� cp � d, respectively.

CHECKING ACCOUNTS 44. The charge for maintaining a checking account at a certain bank is $12 per month plus 10 cents for each check that is written. A competing bank charges $10 per month plus 14 cents per check. Find a criterion for deciding which bank offers

MANUFACTURING OVERHEAD 46. A furniture manufacturer can sell end tables for $70 apiece. It costs the manu- facturer $30 to produce each table, and it is estimated that profit will equal loss when 200 tables are sold. What is the overhead associated with the production of

	the tables? [Note: Overhead is the cost when 0 units are produced.] 47. It costs a book publisher $74,200 to prepare a book for publication (typesetting,

(d) Write an algebraic expression representing the revenue y as a function of the number of books x sold.

(e) Graph both functions on the same coordinate axes.