Q as a linear function of p

i .2 Linear Functions and Applications 17

1.2 Linear Functions and Applications APPLY IT How many units must be sold for a firm to break even?

In Example 6 in this section, this question will be answered using a linear function.

As we saw in the previous section, many situations involve two variables related by a linear equation. For such a relationship, when we express the variable y in terms of x, we say that y is a linear function of x. This means that for any allowed value of x (the independent variable), we can use the equation to find the corresponding value of y (the dependent variable). Exam- ples of equations defining linear functions include y = 2x + 3, y = —5, and 2x — 3y = 7, which can be written as y = (2/3)x — (7/3). Equations in the form x = k, where k is a con- stant, do not define linear functions. All other linear equations define linear functions.

f(x) Notation Letters such as f, g, or h are often used to name functions. For exam- ple, f might be used to name the function defined by

y = 5 — 3x.

To show that this function is named f, it is common to replace y with f (x) (read "f of x") to get

f(x ) =5-3x.

By choosing 2 as a value of x, f (x) becomes 5 — 3 2 = 5 — 6 = —1, written

f(2) = —1.

The corresponding ordered pair is (2, —1). In a similar manner,

f(-4) = 5 — 3(-4) = 17, f(0) = 5, f( —6) = 23,

and so on.

EXAMPLE 1 Function Notation

YOUR TURN 1 Calculate g(-5).

Let g(x) = —4x + 5. Find g(3), g(0), g( —2), and g(b). SOLUTION To find g (3), substitute 3 for x.

g(3) = —4(3) + 5 = —12 + 5 = —7

Similarly,

g(0) = —4( ) + 5 = 0 + 5 = 5,

and

g(-2) = —4( ) + 5 = 8 + 5 = 13,

g(b) = 4 + 5. TRY YOUR TURN 1 IM

We summarize the discussion below.

Linear Function A relationship f defined by

y =f(x)

= mx + b,

for real numbers m and b, is a linear funct ion.

S upply and Demand Linear functions are often good choices for supply and demand curves. Typically, as the price of an item increases, consumers are less likely to buy an increasingly expensive item, and so the demand for the item decreases. On the other

18 CHAPTER I Linear Functions

hand, as the price of an item increases, producers are more likely to see a profit in selling the item, and so the supply of the item increases. The increase in the quantity supplied and decrease in the quantity demanded can eventually result in a surplus, which causes the price to fall. These countervailing trends tend to move the price, as well as the quantity supplied and demanded toward an equilibrium value.

For example, during the late 1980s and early 1990s, the consumer demand for cranber- ries (and all of their healthy benefits) soared. The quantity demanded surpassed the quantity supplied, causing a shortage, and cranberry prices rose dramatically. As prices increased, growers wanted to increase their profits, so they planted more acres of cranberries. Unfortu- nately, cranberries take 3 to 5 years from planting until they can first be harvested. As grow- ers waited and prices increased, consumer demand decreased. When the cranberries were finally harvested, the supply overwhelmed the demand and a huge surplus occurred, caus- ing the price of cranberries to drop in the late 1990s. Source: Agricultural Marketing Resource Center. Other factors were involved in this situation, but the relationship between price, supply, and demand was nonetheless typical.

Although economists consider price to be the independent variable, they have the unfor- tunate habit of plotting price, usually denoted by p, on the vertical axis, while everyone else graphs the independent variable on the horizontal axis. This custom was started by the Eng- lish economist Alfred Marshall (1842-1924). In order to abide by this custom, we will write p, the price, as a function of q, the quantity produced, and plot p on the vertical axis. But remember, it is really price that determines how much consumers demand and produc- ers supply, not the other way around.

Supply and demand functions are not necessarily linear, the simplest kind of function. Yet most functions are approximately linear if a small enough piece of the graph is taken, allowing applied mathematicians to often use linear functions for simplicity. That approach will be taken in this chapter.

EXAMPLE Supply and Demand

Suppose that Greg Tobin, manager of a giant supermarket chain, has studied the supply and demand for watermelons. He has noticed that the demand increases as the price decreases. He has determined that the quantity (in thousands) demanded weekly, q, and the price (in dollars) per watermelon, p, are related by the linear function

p = D(q) = 9 — 0.75q. Demand function

(a) Find the quantity demanded at a price of $5.25 per watermelon and at a price of $3.75 per watermelon. SOLUTION To find the quantity demanded at a price of $5.25 per watermelon, replace p in the demand function with 5.25 and solve for q.

5.25 = 9 — 0.75q

— 3.75 = — 0.75q Subtract 9 from both sides.

5 = q Divide both sides by —0.75.

Thus, at a price of $5.25, the quantity demanded is 5000 watermelons.

Similarly, replace p with 3.75 to find the demand when the price is $3.75. Verify that this leads to q = 7. When the price is lowered from $5.25 to $3.75 per watermelon, the quantity demanded increases from 5000 to 7000 watermelons.

(b) Greg also noticed that the quantity of watermelons supplied decreased as the price decreased. Price p and supply q are related by the linear function

p = 5(q) = 0.75q. Supply function

Find the quantity supplied at a price of $5.25 per watermelon and at a price of $3.00 per watermelon.

u-I /1211 TECHNOLOGY NOTE

p = 9 — 0.75q p = 0.75q (demand) (supply)

(5, 5.25)

(7. 5.25)

(7, 3.75)

p 10

8

6

4

2 4 6 8 10 12 Supply and demand (thousands)

(a)

10

Inters ction ,X=6 Y=45 0

q

0 14

(b) FIGURE II

Linear Functions and Applications 19

YOUR TURN 2 Find the quan- tity of watermelon demanded and supplied at a price of $3.30 per watermelon.

SOLUTION Substitute 5.25 for p in the supply function, p = 0.75q, to find that q = 7, so the quantity supplied is 7000 watermelons. Similarly, replacing p with 3 in the supply equation gives a quantity supplied of 4000 watermelons. If the price decreases from $5.25 to $3.00 per watermelon, the quantity supplied also decreases, from 7000 to 4000 watermelons.

(c) Graph both functions on the same axes. SOLUTION The results of part (a) are written as the ordered pairs (5, 5.25) and (7, 3.75). The line through those points is the graph of the demand function, p = 9 — 0.75q, shown in red in Figure 11(a). We used the ordered pairs (7, 5.25) and (4, 3) from the work in part (b) to graph the supply function, p = 0.75q, shown in blue in Figure 11(a).

TRY YOUR TURN 2 All

A calculator-generated graph of the lines representing the supply and demand functions in Example 2 is shown in Figure 11(b). To get this graph, the equation of each line, using x and y instead of q and p, was entered, along with an appropriate window. A special menu choice gives the coordinates of the intersection point, as shown at the bottom of the graph.

NOTE Not all supply and demand problems will have the same scale on both axes. It helps to consider the intercepts of both the supply graph and the demand graph to decide what scale to use. For example, in Figure 11, the y-intercept of the demand function is 9, so the scale should allow values from 0 to at least 9 on the vertical axis. The x-intercept of the demand function is 12, so the values on the x-axis must go from 0 to 12.

As shown in the graphs of Figure 11, both the supply graph and the demand graph pass through the point (6, 4.5). If the price of a watermelon is more than $4.50, the quantity sup- plied will exceed the quantity demanded and there will be a surplus of watermelons. At a price less than $4.50, the quantity demanded will exceed the quantity supplied and there will be a shortage of watermelons. Only at a price of $4.50 will quantity demanded and supplied be equal. For this reason, $4.50 is called the equilibrium price. When the price is $4.50, quantity demanded and supplied both equal 6000 watermelons, the equilibrium quantity. In general, the equilibrium price of the commodity is the price found at the point where the supply and demand graphs for that commodity intersect. The equilibrium quantity is the quantity demanded and supplied at that same point. Figure 12 illustrates a general supply and demand situation.

p

EXAMPLE 3 Equilibrium Quantity

Use algebra to find the equilibrium quantity and price for the watermelons in Example 2. SOLUTION The equilibrium quantity is found when the prices from both supply and demand are equal. Set the two expressions for p equal to each other and solve.

9 — 0.75q = 0.75q

9 = 1.5q Add 0.75q to both sides.

Supply

Surplus

—Equilibrium

Shortage Demand

6 = q

The equilibrium quantity is 6000 watermelons, the same answer found earlier. The equilibrium price can be found by plugging the value of q = 6 into either the

demand or the supply function. Using the demand function,

p = D(6) = 9 — 0.75(6) = 4.5.

The equilibrium price is $4.50, as we found earlier. Check your work by also plugging q = 6 into the supply function. TRY YOUR TURN 3 Al

q

FIGURE 12

YOUR TURN 3 Repeat Example 3 using the demand equation D(q) = 10 — 0.85q and the supply equation S(q) = 0.4q. INF

20 CHAPTER I Linear Functions

YAM TECHNOLOGY NOTE You may prefer to find the equilibrium quantity by solving the equation with your calculator. Or, if your calculator has a TABLE feature, you can use it to find the value of q that makes the two expres-sions equal.

Another important issue is how, in practice, the equations of the supply and demand functions can be found. Data need to be collected, and if they lie perfectly along a line, then the equation can easily be found with any two points. What usually happens, however, is that the data are scattered, and there is no line that goes through all the points. In this case we must find a line that approximates the linear trend of the data as closely as possible (assuming the points lie approximately along a line) as in Example 14 in the previous sec- tion. This is usually done by the method of least squares, also referred to as linear regres- sion. We will discuss this method in Section 1.3.

Cost Analysis The cost of manufacturing an item commonly consists of two parts. The first is a tied cost for designing the product, setting up a factory, training workers, and so on. Within broad limits, the fixed cost is constant for a particular product and does not change as more items are made. The second part is a cost per item for labor, materials, packing, shipping, and so on. The total value of this second cost does depend on the num- ber of items made.

EXAMPLE 4' Cost Analysis

A small company decides to produce video games. The owners find that the fixed cost for creating the game is $5000, after which they must spend $12 to produce each individual copy of the game. Find a formula C(x) for the cost as a linear function of x, the number of games produced. SOLUTION Notice that C(0) = 5000, since $5000 must be spent even if no games are produced. Also, C(1) = 5000 + 12 = 5012, and C(2) = 5000 + 2 • 12 = 5024. In general,

C(x) = 5000 + 12x,

because every time x increases by 1, the cost should increase by $12. The number 12 is also the slope of the graph of the cost function; the slope gives us the cost to produce one addi- tional item.

In economics, marginal cost is the rate of change of cost C(x) at a level of production x and is equal to the slope of the cost function at x. It approximates the cost of producing one additional item. In fact, some books define the marginal cost to be the cost of producing one additional item. With linear functions, these two definitions are equivalent, and the mar- ginal cost, which is equal to the slope of the cost function, is constant. For instance, in the video game example, the marginal cost of each game is $12. For other types of functions, these two definitions are only approximately equal. Marginal cost is important to manage- ment in making decisions in areas such as cost control, pricing, and production planning.

The work in Example 4 can be generalized. Suppose the total cost to make x items is given by the linear cost function C(x) = mx + b. The fixed cost is found by letting x = 0:

c(o) = m • 0 + = b;

thus, the fixed cost is b dollars. The additional cost of each additional item, the marginal cost, is m, the slope of the line C(x) = mx + b.

1000 900 800 700 600 500 400 300 200 100

C(x) = 20x + 100

-even point

R(x) =

0 5 10 15 20 25 30 35 40 45 50

Units of feed

FIGURE 13

I.2 Linear Functions and Applications 21

Linear Cost Function In a cost function of the form C(x) = mx + b, the m represents the marginal cost and b the fixed cost. Conversely, if the fixed cost of producing an item is b and the marginal cost is m, then the linear cost function C(x) for producing x items is C(x) = mx + b.

EXAMPLE

Cost Function

YOUR TURN 4 Repeat Exam- ple 5, using a marginal cost of $15 per batch and a cost of $1930 to produce 80 batches.

The marginal cost to make x batches of a prescription medication is $10 per batch, while the cost to produce 100 batches is $1500. Find the cost function C(x), given that it is linear. SOLUTION Since the cost function is linear, it can be expressed in the form C(x) = mx + b. The marginal cost is $10 per batch, which gives the value for m. Using x = 100 and C(x) = 1500 in the point-slope form of the line gives

C(x) — 1500 = 10(x — 100)

C(x) — 1500 = 10x — 1000

C(x) = 10x + 500. Add 1500 to both sides.

The cost function is given by C(x) = 10x + 500, where the fixed cost is $500. TRY YOUR TURN 4

Break-Even Analysis The revenue R(x) from selling x units of an item is the product of the price per unit p and the number of units sold (demand) x, so that

R(x) = px.

The corresponding profit P(x) is the difference between revenue R(x) and cost C(x). That is,

P(x) = R(x) — C(x).

A company can make a profit only if the revenue received from its customers exceeds the cost of producing and selling its goods and services. The number of units at which revenue just equals cost is the break-even quantity; the corresponding ordered pair gives the break-even point.

EXAMPLE 6 Break-Even Analysis

OW APPLY IT A firm producing poultry feed finds that the total cost C(x) in dollars of producing and sell- ing x units is given by

C(x) = 20x + 100.

Management plans to charge $24 per unit for the feed.

(a) How many units must be sold for the firm to break even? SOLUTION The firm will break even (no profit and no loss) as long as revenue just equals cost, or R(x) = C(x). From the given information, since R(x) = px and p = $24,

R(x) = 24x.

Substituting for R(x) and C(x) in the equation R(x) = C(x) gives

24x = 20x + 100,

from which x = 25. The firm breaks even by selling 25 units, which is the break-even quantity. The graphs of C(x) = 20x + 100 and R(x) = 24x are shown in Figure 13.

22 CHAPTER I Linear Functions

The break-even point (where x = 25) is shown on the graph. If the company sells more than 25 units (if x > 25), it makes a profit. If it sells fewer than 25 units, it loses money.

(b) What is the profit if 100 units of feed are sold? SOLUTION Use the formula for profit P(x).

P(x) = R(x) — C(x) = 24x — (20x + 100) = 4x — 100

Then P(100) = 4(100) — 100 = 300. The firm will make a profit of $300 from the sale of 100 units of feed.

(c) How many units must be sold to produce a profit of $900? SOLUTION Let P(x) = 900 in the equation P(x) = 4x — 100 and solve for x.

900 = 4x — 100 1000 = 4x

X = 250

Sales of 250 units will produce $900 profit. TRY YOUR TURN 5 la

Temperature One of the most common linear relationships found in everyday situa- tions deals with temperature. Recall that water freezes at 32° Fahrenheit and 0° Celsius, while it boils at 212° Fahrenheit and 100° Celsius.* The ordered pairs (0, 32) and (100, 212) are graphed in Figure 14 on axes showing Fahrenheit (F) as a function of Cel-

sius (C). The line joining them is the graph of the function.

FIGURE 14

Temperature

Derive an equation relating F and C.

SOLUTION To derive the required linear equation, first find the slope using the given

ordered pairs, (0, 32) and (100, 212).

212 — 32 9 m 100 — 0 5

*Gabriel Fahrenheit (1686-1736), a German physicist, invented his scale with 0° representing the temperature of an equal mixture of ice and ammonium chloride (a type of salt), and 96° as the temperature of the human body. (It is often said, erroneously, that Fahrenheit set 100° as the temperature of the human body. Fahrenheit's own

words are quoted in A History of the Thermometer and Its Use in Meteorology by W. E. Knowles, Middleton: The

Johns Hopkins Press, 1966, p. 75.) The Swedish astronomer Anders Celsius (1701-1744) set 0° and 100° as the

freezing and boiling points of water.

YOUR TURN 5 Repeat Exam- ple 6(c), using a cost function C(x) = 35x + 250, a charge of

$18 per unit, and a profit of $8030,

EXAMPLE 7

1.2 Linear Functions and Applications 23

The F-intercept of the graph is 32, so by the slope-intercept form, the equation of the line is

9 F = -

5 C + 32.

With simple algebra this equation can be rewritten to give C in terms of F:

C = 9 - 5

(F - 32).

I.2 EXERCISES For Exercises 1-10, let f(x) = 7 - 5x and g(x) = 2x - 3. Find the following.

1. f (2) 2. f (4)

3. f(- 3) 4. f(-1) 5. g(1.5) 6. g(2.5)

7. g (- 1/2) 8. g ( -314)

9. f (t) 10. g (k2)

In Exercises 11-14, decide whether the statement is true or false.

11. To find the x-intercept of the graph of a linear function, we solve y = f (x) = 0, and to find the y-intercept, we evaluate f (0).

12. The graph of f (x) = -5 is a vertical line.

13. The slope of the graph of a linear function cannot be undefined.

14. The graph of f(x) = ax is a straight line that passes through the origin. Describe what fixed costs and marginal costs mean to a company.

.7S.16. In a few sentences, explain why the price of a commodity not already at its equilibrium price should move in that direction.

_17. Explain why a linear function may not be adequate for describ- ing the supply and demand functions.

\18. In your own words, describe the break-even quantity, how to find it, and what it indicates.

Write a linear cost function for each situation. Identify all vari- ables used.

19. A Lake Tahoe resort charges a snowboard rental fee of $10 plus $2.25 per hour.

20. An Internet site for downloading music charges a $10 registra- tion fee plus 99 cents per downloaded song.

21. A parking garage charges 2 dollars plus 75 cents per half-hour.

22. For a one-day rental, a car rental firm charges $44 plus 28 cents per mile.

Assume that each situation can be expressed as a linear cost function. Find the cost function in each case.

23. Fixed cost: $100; 50 items cost $1600 to produce.

24. Fixed cost: $35; 8 items cost $395 to produce.

25. Marginal cost: $75; 50 items cost $4300 to produce.

26. Marginal cost: $120; 700 items cost $96,500 to produce.

APPLICATIONS Business and Economics 27. Supply and Demand Suppose that the demand and price for

a certain model of a youth wristwatch are related by

p = D(q) = 16 - 1.25q,

where p is the price (in dollars) and q is the quantity demanded (in hundreds). Find the price at each level of demand.

a. 0 watches b. 400 watches c. 800 watches Find the quantity demanded for the watch at each price.

d. $8 e. $10 f. $12 g. Graph p = 16 - 1.25q. Suppose the price and supply of the watch are related by

p = S(q) = 0.75q,

where p is the price (in dollars) and q is the quantity supplied (in hundreds) of watches. Find the quantity supplied at each price.

h. $0 i. $10 j. $20

k. Graph p = 0.75q on the same axis used for part g.

I. Find the equilibrium quantity and the equilibrium price.

28. Supply and Demand Suppose that the demand and price for strawberries are related by

p = D(q) = 5 - 0.25q,

where p is the price (in dollars) and q is the quantity demanded (in hundreds of quarts). Find the price at each level of demand.

a. 0 quarts b. 400 quarts c. 840 quarts

24 CHAPTER I Linear Functions

Find the quantity demanded for the strawberries at each pric e.

d. $4.50 e. $3.25 f. $2.40

g. Graph p = 5 — 0.25q.

Suppose the price and supply of strawberries are related by

p = 5(q) = 0.25q,

where p is the price (in dollars) and q is the quantity supplied (in hundreds of quarts) of strawberries. Find the quantity supplied at each price.

h. $0 I. $2

j. $4.50

k. Graph p = 0.75q on the same axis used for part g.

1. Find the equilibrium quantity and the equilibrium price.

29. Supply and Demand Let the supply and demand functions for butter pecan ice cream be given by

p = S(g) = q and p = D (q) = 100 —

where p is the price in dollars and q is the number of 10-gallon tubs.

a. Graph these on the same axes.

b. Find the equilibrium quanti ty and the equilibrium price. (Hint: The way to divide by a fraction is to multiply by its reciprocal.)

30. Supply and Demand Let the supply and demand functions for sugar be given by

p = S(q) = 1.4q — 0.6 and

p = D( q) = —2q + 3.2,

where p is the price per pound and g is the quantity in thou- sands of pounds.

a. Graph these on the same axes.

b. Find the equilibrium quantity and the equilibrium price.

31. Supply and Demand Suppose that the supply function for honey is p = S(q) = 0.3q + 2.7, where p is the price in dollars for an 8-oz container and q is the quantity in barrels. Suppose also that the equilibrium price is $4.50 and the demand is 2 barrels when the price is $6.10. Find an equation for the demand function, assuming it is linear.

32. Supply and Demand Suppose that the supply function for walnuts is p = S(q) = 0.25q + 3.6, where p is the price in dollars per pound and q is the quantity in bushels. Suppose also that the equilibrium price is $5.85, and the demand is 4 bushels when the price is $7.60. Find an equation for the demand function, assuming it is linear.

33. T-Shirt Cost Joanne Wendelken sells silk-screened T-shirts at community festivals and crafts fairs. Her marginal cost to pro- duce one T-shirt is $3.50. Her total cost to produce 60 T-shirts is $300, and she sells them for $9 each.

a. Find the linear cost function for Joanne's T-shirt production.

b. How many T-shirts must she produce and sell in order to break even?

c. How many T-shirts must she produce and sell to make a profit of $500?

34. Publishing Costs Alfred Juarez owns a small publishing house specializing in Latin American poetry. His fixed cost to produce a typical poetry volume is $525, and his total cost to produce 1000 copies of the book is $2675. His books sell for $4.95 each. a. Find the linear cost function for Alfred's book production.

b. How many poetry books must he produce and sell in order to break even?

c. How many books must he produce and sell to make a profit of $1000?

35. Marginal Cost of Coffee The manager of a restaurant found that the cost to produce 100 cups of coffee is $11.02, while the cost to produce 400 cups is $40.12. Assume the cost C(x) is a linear function of x, the number of cups produced. a. Find a formula for C(x).

b. What is the fixed cost?

c. Find the total cost of producing 1000 cups.

d. Find the total cost of producing 1001 cups.

e. Find the marginal cost of the 1001st cup.

f. What is the marginal cost of any cup and what does this mean to the manager?

36. Marginal Cost of a New Plant In deciding whether to set up a new manufacturing plant, company analysts have decided that a linear function is a reasonable estimation for the total cost C(x) in dollars to produce x items. They estimate the cost to produce 10,000 items as $547,500, and the cost to produce 50,000 items as $737,500.

a. Find a formula for C(x).

b. Find the fixed cost.

c. Find the total cost to produce 100,000 items.

d. Find the marginal cost of the items to be produced in this plant and what does this mean to the manager?

37. Break-Even Analysis Producing x units of tacos costs C(x) = 5x + 20; revenue is R(x) = 15x, where C(x) and R(x) are in dollars.

a. What is the break-even quantity?

b. What is the profit from 100 units?

c. How many units will produce a profit of $500?

38. Break-Even Analysis To produce x units of a religious medal costs C(x) = 12x + 39. The revenue is R(x) = 25x. Both C(x) and R(x) are in dollars.

a. Find the break-even quantity.

b. Find the profit from 250 units.

c. Find the number of units that must be produced for a profit of $130.

Break-Even Analysis You are the manager of a firm. You are considering the manufacture of a new product, so you ask the accounting department for cost estimates and the sales depart- ment for sales estimates. After you receive the data, you must decide whether to go ahead with production of the new prod- uct. Analyze the data in Exercises 39-42 (find a break-even

1.3 The Least Squares Line 25

quantity) and then decide what you would do in each case. Also write the profit function.

39. C(x) = 85x + 900; R(x) = 105x; no more than 38 units can be sold.

40. C(x) = 105x + 6000; R(x) = 250x; no more than 400 units can be sold.

41. C(x) = 70x + 500; R(x) = 60x (Hint: What does a negative break-even quantity mean?)

42. C(x) = 1000x + 5000; R(x) = 900x

43. Break-Even Analysis Suppose that the fixed cost for a prod- uct is $400 and the break-even quantity is 80. Find the mar- ginal profit (the slope of the linear profit function).

44. Break-Even Analysis Suppose that the fixed cost for a prod- uct is $650 and the break-even quantity is 25. Find the mar- ginal profit (the slope of the linear profit function).

Physical Sciences

45. Temperature Use the formula for conversion between Fahrenheit and Celsius derived in Example 7 to convert each temperature.

a. 58°F to Celsius

b. -20°F to Celsius

c. 50°C to Fahrenheit

46. Body Temperature You may have heard that the average tem- perature of the human body is 98.6°. Recent experiments show that the actual figure is closer to 98.2°. The figure of 98.6 comes from experiments done by Carl Wunderlich in 1868. But

Wunderlich measured the temperatures in degrees Celsius and rounded the average to the nearest degree, giving 37°C as the average temperature. Source: Science News.

a. What is the Fahrenheit equivalent of 37°C?

b. Given that Wunderlich rounded to the nearest degree Cel- sius, his experiments tell us that the actual average human body temperature is somewhere between 36.5°C and 37.5°C. Find what this range corresponds to in degrees Fahrenheit.

47. Temperature Find the temperature at which the Celsius and Fahrenheit temperatures are numerically equal.

General Interest

48. Education Cost The 2009-2010 budget for the California State University system projected a fixed cost of $486,000 at each of five off-campus centers, plus a marginal cost of $1140 per student. Source: California State University.

a. Find a formula for the cost at each center, C(x ), as a linear function of x, the number of students.

b. The budget projected 500 students at each center. Calculate the total cost at each center.

c. Suppose, due to budget cuts, that each center is limited to $1 million. What is the maximum number of students that each center can then support?

Mr YOUR TURN ANSWERS

1. 25 2. 7600 and 4400 3. 8000 watermelons and $3.20 per watermelon 4. C(x) = 15x + 730 5. 360

1.3 The Least Squares Line APPLY IT

Accidental Death Rate

Year Death Rate 1910 84.4

71.2 1920 1930 80.5 1940 73.4 1950 60.3 1960 52.1 1970 56.2 1980 46.5 1990 36.9 2000 34.0

How has the accidental death rate in the United States changed over time? In Example 1 in this section, we show how to answer such questions using the method of least squares.

We use past data to find trends and to make tentative predictions about the future. The only assumption we make is that the data are related linearly-that is, if we plot pairs of data, the resulting points will lie close to some line. This method cannot give exact answers. The best we can expect is that, if we are careful, we will get a reasonable approximation.

The table lists the number of accidental deaths per 100,000 people in the United States through the past century. Source: National Center for Health Statistics. If you were a man- ager at an insurance company, these data could be very important. You might need to make some predictions about how much you will pay out next year in accidental death benefits, and even a very tentative prediction based on past trends is better than no prediction at all.