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118 2 LINEAR AND QUADRATIC FUNCTIONS

71. Celsius/Fahrenheit. A formula for converting Celsius de-

grees to Fahrenheit degrees is given by the linear function

F 32

Determine to the nearest degree the Celsius range in tem-

perature that corresponds to the Fahrenheit range of 60°F

to 80°F.

72. Celsius/Fahrenheit. A formula for converting Fahrenheitdegrees to Celsius degrees is given by the linear function

C (F 32)

Determine to the nearest degree the Fahrenheit range in

temperature that corresponds to a Celsius range of 20°C to

30°C.

73. Earth Science. In 1984, the Soviets led the world in

drilling the deepest hole in the Earth’s crust—more than

12 kilometers deep. They found that below 3 kilometers

the temperature T increased 2.5°C for each additional 100

meters of depth.

(A) If the temperature at 3 kilometers is 30°C and x is the

depth of the hole in kilometers, write an equation

using x that will give the temperature T in the hole at

any depth beyond 3 kilometers.

(B) What would the temperature be at 15 kilometers?

[The temperature limit for their drilling equipment

was about 300°C.]

5

9

9

5C

(C) At what interval of depths will the temperature be

between 200°C and 300°C, inclusive?

74. Aeronautics. Because air is not as dense at high altitudes,

planes require a higher ground speed to become airborne.

A rule of thumb is 3% more ground speed per 1,000 feet of

elevation, assuming no wind and no change in air tempera-

ture. (Compute numerical answers to 3 significant digits.)

(A) Let

V s Takeoff ground speed at sea level for a particular

plane (in miles per hour)

A Altitude above sea level (in thousands of feet)

V Takeoff ground speed at altitude A for the same

plane (in miles per hour)

Write a formula relating these three quantities.

(B) What takeoff ground speed would be required at Lake

Tahoe airport (6,400 feet), if takeoff ground speed at

San Francisco airport (sea level) is 120 miles per

hour?

(C) If a landing strip at a Colorado Rockies hunting lodge(8,500 feet) requires a takeoff ground speed of 125

miles per hour, what would be the takeoff ground

speed in Los Angeles (sea level)?

(D) If the takeoff ground speed at sea level is 135 miles

per hour and the takeoff ground speed at a mountain

resort is 155 miles per hour, what is the altitude of the

mountain resort in thousands of feet?

Section 2-3 Quadratic Functions

Quadratic Functions

Completing the Square

Properties of Quadratic Functions and Their Graphs

Applications

Quadratic Functions

The graph of the square function, h( x) x2, is shown in Figure 1. Notice that thegraph is symmetric with respect to the y axis and that (0, 0) is the lowest pointon the graph. Let’s explore the effect of applying a sequence of basic transfor-

mations to the graph of h. (A brief review of Section 1-5 might prove helpful atthis point.)

h(x )

5

5 5x

FIGURE 1Square function h( x) x2.



2-3 Quadratic Functions

Explore/Discuss

1

Indicate how the graph of each function is related to the graph ofh( x) x2. Discuss the symmetry of the graphs and find the highest orlowest point, whichever exists, on each graph.

(A) f ( x) ( x 3)2 7 x2 6 x 2

(B) g( x) 0.5( x 2)2 3 0.5 x2 2 x 5

(C) m( x) ( x 4)2

8 x2

8 x 8(D) n( x) 3( x 1)2 1 3 x2 6 x 4

Graphing the functions in Explore/Discuss 1 produces figures similar in shto the graph of the square function in Figure 1. These figures are called para

las. The functions that produced these parabolas are examples of the imporclass of quadratic functions, which we now define.

QUADRATIC FUNCTIONS

If a, b, and c are real numbers with a 0, then the function

f ( x) ax2 bx c

is a quadratic function and its graph is a parabola.*

Since the expression ax2 bx c represents a real number for all real nber replacements of x,

the domain of a quadratic function is the set of all real numbers.

We will discuss methods for determining the range of a quadratic function lin this section. Typical graphs of quadratic functions are illustrated in Figure

(a) f ( x ) x 2 9 (b) g( x ) 2 x 2 15 x 30 (c) h( x ) 0.3 x 2 x 4

Completing the SquareIn Explore/Discuss 1 we wrote each function as two different, but equivalexpressions. For example,

f ( x) ( x 3)2 7 x2 6 x 2

10

10

10

10

10

10

10

10

10

10

10

FIGURE 2Graphs of quadratic

functions.

*A more general definition of a parabola that is independent of any coordinate system is give

Section 7-1.



Explore/Discuss

2


It is easy to verify that these two expressions are equivalent by expanding thefirst expression. The first expression is more useful than the second for analyzingthe graph of f. If we are given only the second expression, how can we determinethe first? It turns out that this is a routine process, called completing the square,that is another useful tool to be added to our mathematical toolbox.

Replace ? in each of the following with a number that makes theequation valid.

(A) ( x 1)2 x2 2 x ? (B) ( x 2)2 x2 4 x ?

(C) ( x 3)2 x2 6 x ? (D) ( x 4)2 x2 8 x ?

Replace ? in each of the following with a number that makes the expres-sion a perfect square of the form ( x h)2.

(E) x2 10 x ? (F) x2

12 x ? (G) x2 bx ?

Given the quadratic expression

x2 bx

what must be added to this expression to make it a perfect square? To find out,consider the square of the following expression:

We see that the third term on the right side of the equation is the square of one-half the coefficient of x in the second term on the right; that is, m2 is the squareof (2m). This observation leads to the following rule:

COMPLETING THE SQUARE

To complete the square of the quadratic expression

x2 bx

add the square of one-half the coefficient of x; that is, add

or

The resulting expression can be factored as a perfect square:

x2

bx b22

x b

22

b2

4b22

1

2

m2 is the square of one-half thecoefficient of x.

( x m)2 x 2 2mx m

2




Completing the Square

Complete the square for each of the following:

(A) x2 3 x (B) x2 6bx

S o l u t i o n s (A) x2 3 x

x2

3 x Add that is,

(B) x2 6bx

x2 6bx 9b2

( x 3b)2 Add that is, 9b2.

Complete the square for each of the following:

(A) x2 5 x (B) x2

4mx

It is important to note that the rule for completing the square applies to oquadratic expressions in which the coefficient of x2 is 1. This causes little trble, however, as you will see.

Properties of Quadratic Functions and Their Graphs

We now use the process of completing the square to transform the quadratic func

f ( x) ax2 bx c

into the standard form

f ( x) a( x h)2 k

Many important features of the graph of a quadratic function can be determiby examining the standard form. We begin with a specific example and then geralize the results.

Consider the quadratic function given by

f ( x) 2 x2 8 x 4

We use completing the square to transform this function into standard form:

f ( x) 2 x2 8 x 4

2( x2 4 x) 4

2( x2 4 x ?) 4

2( x2 4 x 4) 4 8 We add 4 to complete the squareinside the parentheses. But becauof the 2 outside the parentheses,we have actually added 8, so wemust subtract 8.

2( x 2)2 4 The transformation is complete acan be checked by expanding.

Factor the coefficient of x 2 out o

the first two terms.

M A T C H E D P R O B L E M

1

6b

2 2

;

9

4.3

2 2

;

x

3

22

9

4

E X A M P L E

1




Thus, the standard form is

f ( x) 2( x 2)2 4 (2)

If x 2, then 2( x 2)2 0 and f (2) 4. For any other value of x, thepositive number 2( x 2)2 is added to 4, making f ( x) larger. Therefore,

f (2)

4

is the minimum value of f ( x) for all x—a very important result! Furthermore, if we choose any two values of x that are equidistant from x 2, we will obtainthe same value for the function. For example, x 1 and x 3 are each one unitfrom x 2 and their functional values are

f (1) 2(1)2 4 2

f (3) 2(1)2 4 2

Thus, the vertical line x 2 is a line of symmetry—if the graph of equation (1)is drawn on a piece of paper and the paper folded along the line x 2, then the

two sides of the parabola will match exactly.The above results are illustrated by graphing equation (1) or (2) and the line

x 2 in a suitable viewing window (Fig. 3).

From the analysis of equation (2), illustrated by the graph in Figure 3, we con-clude that f ( x) is decreasing on (, 2] and increasing on [2, ). Furthermore,

f ( x) can assume any value greater than or equal to 4, but no values less than4. Thus,

Range of f: y 4 or [4, )

In general, the graph of a quadratic function is a parabola with line of sym-metry parallel to the vertical axis. The lowest or highest point on the parabola,whichever exists, is called the vertex. The maximum or minimum value ofa quadratic function always occurs at the vertex of the graph. The verticalline of symmetry through the vertex is called the axis of the parabola. Thus,for f ( x) 2 x2 8 x 4, the vertical line x 2 is the axis of the parabola and(2, 4) is its vertex.

10

4

10

6

f (x ) 2x 2 8x 4 2(x 2)2 4

Axis of symmetry:x 2

Minimum:f (2) 4

FIGURE 3Graph of a quadratic function.



Explore/Discuss


From equation (2), we can see that the graph of f is simply the graphg( x) 2 x2 translated to the right 2 units and down 4 units, as shown in Figur

Notice the important results we have obtained from the standard form of quadratic function f:

The vertex of the parabola

The axis of the parabola

The minimum value of f ( x)

The range of f

A relationship between the graph of f and the graph of g

Explore the effect of changing the constants a, h, and k on the graph of f ( x) a( x h)2 k.

(A) Let a 1 and h 5. Graph function f for k 4, 0, and 3simultaneously in the same viewing window. Explain the effect of changing k on the graph of f.

(B) Let a 1 and k 2. Graph function f for h 4, 0, and 5simultaneously in the same viewing window. Explain the effect of changing h on the graph of f.

(C) Let h 5 and k 2. Graph function f for a 0.25, 1, and 3simultaneously in the same viewing window. Graph function f for a

1, 1, and 0.25 simultaneously in the same viewing window.Explain the effect of changing a on the graph of f.

(D) Can all quadratic functions of the form y ax2 bx c be

rewritten as a( x h)2

k ?

We generalize the above discussion in the following box:

10

4

10

6

f (x ) 2x 2 8x 4 2(x 2)2 4

g (x ) 2x 2FIGURE 4Graph of f is the graph of g

translated.

3




PROPERTIES OF A QUADRATIC FUNCTION AND ITS GRAPH

Given a quadratic function and the standard form obtained by completingthe square

f ( x) ax2 bx c a( x h)2 k a 0

we summarize general properties as follows:1. The graph of f is a parabola:

2. Vertex: (h, k ) (parabola increases on one side of the vertex anddecreases on the other).

3. Axis (of symmetry): x h (parallel to y axis).

4. f (h) k is the minimum if a 0 and the maximum if a 0.

5. Domain: all real numbers; range: (, k ] if a 0 or [k , ) ifa 0.

6. The graph of f is the graph of g( x) ax2 translated horizontally h

units and vertically k units.

Analyzing a Quadratic Function

Find the standard form for the following quadratic function, analyze the graph,and check your results with a graphing utility:

f ( x) 0.5 x2 x 5

S o l u t i o n We complete the square to find the standard form:

f ( x) 0.5 x2 x 5

0.5( x2 2 x ?) 5

0.5( x2 2 x 1) 5 0.5

0.5( x 1)2 5.5

E X A M P L E

2

x

f (x )

k

h

Axisx h

Vertex (h, k )

Max f (x )

a 0Opens downward

x

f (x )

k

h

Axisx h

Vertex (h, k )

Min f (x )

a 0Opens upward




From the standard form we see that h 1 and k 5.5. Thus, the verte(1, 5.5), the axis of symmetry is x 1, the maximum value is f (1)

and the range is (, 5.5]. The function f is increasing on (, 1] and decring on [1, ). The graph of f is the graph of g( x) 0.5 x2 shifted to the 1 unit and upward 5.5 units. To check these results, we graph f and g simultaously in the same viewing window, use the built-in maximum routine to lothe vertex, and add the graph of the axis of symmetry (Fig. 5).

Find the standard form for the following quadratic function, analyze the graand check your results with a graphing utility:

f ( x) x2 3 x 1

Finding the Equation of a Parabola

Find an equation for the parabola whose graph is shown in Figure 6.

(a) (b)

S o l u t i o n Figure 6(a) shows that the vertex of the parabola is (h, k ) (3, 2). Thus,standard equation must have the form

f ( x) a( x 3)2 2

Figure 6(b) shows that f (4) 0. Substituting in equation (3) and solving fowe have

f (4) a(4 3)2 2 0

a 2

Thus, the equation for the parabola is

f ( x) 2( x 3)2 2 2 x2 12 x 16

Find the equation of the parabola with vertex (2, 4) and y intercept (0, 2).M A T C H E D P R O B L E M

3

5

0

5

6

5

0

5

6

FIGURE 6

E X A M P L E

3


2

6

6

6

6

FIGURE 5




Applications

We now look at several applications that can be modeled using quadratic functions.

Maximum Area

A dairy farm has a barn that is 150 feet long and 75 feet wide. The owner has

240 ft of fencing and wishes to use all of it in the construction of two iden-tical adjacent outdoor pens with the long side of the barn as one side of thepens and a common fence between the two (Fig. 7). The owner wants the pensto be as large as possible.

(A) Construct a mathematical model for the combined area of both pens inthe form of a function A( x) (see Fig. 7) and state the domain of A.

(B) Find the value of x that produces the maximum combined area.

(C) Find the dimensions and the area of each pen.

S o l u t i o n s (A) Since y 240 3 x,

A( x) (240 3 x) x 240 x 3 x2

The distances x and y must be nonnegative. Since y 240 3 x, it followsthat x cannot exceed 80. Thus, a model for this problem is

A( x) 240 x 3 x2, 0 x 80

(B) Omitting the details, the standard form for A is

A( x) 3( x 40)2 4,800

Thus, the maximum combined area of 4,800 ft2 occurs at x 40. This resultis confirmed in Figure 8.

(C) Each pen is x by y /2 or 40 ft by 60 ft. The area of each pen is 40 ft

60 ft 2,400 ft2.

x

x y

x

150 feet

75 feet

FIGURE 7

E X A M P L E

4

0

0

5,000

80

FIGURE 8 A( x) 240 x 3 x2.




Repeat Example 4 with the owner constructing three identical adjacent pinstead of two.

Now that we have added quadratic functions to our mathematical toolbox,can use this new tool in conjunction with another tool discussed previouslyregression analysis. In the next example, we use both of these tools to investigthe effect of recycling efforts on solid waste disposal.

Solid Waste Disposal

Franklin Associates Ltd. of Prairie Village, Kansas, reported the data in Ta1 to the U.S. Environmental Protection Agency.

(A) Let x represent time in years with x 0 corresponding to 1960, and y represent the corresponding annual landfill disposal. Use regresanalysis on a graphing utility to find a quadratic function of the fo

y ax2 bx c that models this data. (Round the constants a, b,

c to three significant digits* when reporting your results.)

(B) If landfill disposal continues to follow the trend exhibited in Tablwhen (to the nearest year) would the annual landfill disposal return to1970 level?

(C) Is it reasonable to expect the annual landfill disposal to follow this trindefinitely? Explain.

S o l u t i o n s (A) Since the values of y increase from 1970 to 1987 and then begin to decre

a quadratic model seems a better choice than a linear one. Figure 9 shothe details of constructing the model on a graphing utility.

T A B L E 1 Municipal Solid Waste Disposal

Year1970

1980

1985

1987

1990

1993

1995

Per Person Per

(pounds)

Annual Landfill Disposal

(millions of tons) 88.2

123.3

136.4

140.0

131.6

127.6

118.4

2.37

2.97

3.13

3.15

2.90

2.70

2.50

E X A M P L E

5


4

*For those not familiar with the meaning of significant digits, see Appendix C for a brief discussio

this concept.




Rounding the constants to three significant digits, a quadratic regressionequation for this data is

y1 0.187 x2 9.77 x 7.99

The graph in Figure 9(d) indicates that this is a reasonable model for thisdata. It is, in fact, the “best” quadratic equation for this data.

(B) To determine when the annual landfill disposal returns to the 1970 level, we

add the graph of y2 88.2 to the graph [Fig. 10(a)]. The graphs of y1 and y2 intersect twice, once at x 10 (1970), and again at a later date. Using abuilt-in intersection routine [Fig. 10(b)] shows that the x coordinate of thesecond intersection point (to the nearest integer) is 42. Thus, the annual land-fill disposal returns to the 1970 level of 88.2 million tons in 2002. [ Note:

You will obtain slightly different results if you round the constants a, b, andc before finding the intersection point. As we stated before, we will alwaysuse the unrounded constants in calculations and only round the final answer.]

(a) (b)

(C) The graph of y1 continues to decrease and reaches 0 somewhere between2110 and 2115. It is highly unlikely that the annual landfill disposal will everreach 0. As time goes by and more data becomes available, new models willhave to be constructed to better predict future trends.

Refer to Table 1.

(A) Let x represent time in years with x 0 corresponding to 1960, and let y

represent the corresponding landfill disposal per person per day. Use regres-sion analysis on a graphing utility to find a quadratic function of the form

y ax2 bx c that models this data. (Round the constants a, b, and c

to three significant digits when reporting your results.)


5

0

0

150

60

0

0

150

60

y 2 88.2FIGURE 10

(d) Graph of data andregression equation

(c) Regression equationtransferred to equationeditor

(b) Regression equation(a) Data

FIGURE 9

0

0

150

60



(B) If landfill disposal per person per day continues to follow the trend exited in Table 1, when (to the nearest year) would it fall below 1.5 pouper person per day?

(C) Is it reasonable to expect the landfill disposal per person per day to folthis trend indefinitely? Explain.

Answers t o Matched P rob lems

1. (A) (B) x2 4mx 4m2 ( x 2m)2

2. Standard form: f ( x) ( x 1.5)2 1.25. The vertex is (1.5, 1.25), the axis of symmetry is x 1.5, the maximum valu

of f ( x) is 1.25, and the range of f is (, 1.25]. The function f is increasing on (, 1.5] and decreasing on [1.5, ). Th

graph of f is the graph of g( x) x2 shifted 1.5 units to the right and 1.25 units upward.

3. f ( x) 0.5( x 2)2 4 0.5 x2 2 x 2

4. (A) A( x) (240 4 x) x, 0 x 60 (B) The maximum combined area of 3,600 ft2 occurs at x 30 ft.

(C) Each pen is 30 ft by 40 ft with area 1,200 ft2.

5. (A) y 0.00434 x2 0.202 x 0.759 (B) 2003

x2

5 x 25

4 x

5

22

EXERCISE 2-3

A

In Problems 1–6, complete the square and find the standard

form of each quadratic function.

1. f ( x) x2 4 x 5 2. g( x) x2 2 x 3

3. h( x) x2 2 x 1 4. k ( x) x2 4 x 4

5. m( x) x2 4 x 1 6. n( x) x2 2 x 3

In Problems 7 –12, write a brief verbal description of the

relationship between the graph of the indicated function (from

Problems 1–6) and the graph of y x2.

7. f ( x) x2 4 x 5 8. g( x) x2 2 x 3

9. h( x) x2 2 x 1 10. k ( x) x2 4 x 4

11. m( x) x2 4 x 1 12. n( x) x2 2 x 3

In Problems 13–18, match each graph with one of the

functions in Problems 1–6.

13.

5

5

5

5

14.

15.

16.

17.

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5





18.

B

For each quadratic function in Problems 19–24, sketch a

graph of the function and label the axis and the vertex.

19. f ( x) 2 x2 24 x 90 20. f ( x) 3 x2 24 x 30

21. f ( x) x2 6 x 4 22. f ( x) x2 10 x 30

23. f ( x) 0.5 x2 2 x 7 24. f ( x) 0.4 x2 4 x 4

In Problems 25–28, find the intervals where f is increasing, the

intervals where f is decreasing, and the range. Expressanswers in interval notation.

25. f ( x) 4 x2 18 x 25

26. f ( x) 5 x2 29 x 17

27. f ( x) 10 x2 44 x 12

28. f ( x) 8 x2 20 x 16

In Problems 29–32, use the graph of the parabola to find the

equation of the corresponding quadratic function.

29.

30.

31.

5

5

5

5

32.

In Problems 33–

38, find the equation of a quadratic functionwhose graph satisfies the given conditions.

33. Vertex: (4, 8); x intercept: 6

34. Vertex: (2, 12); x intercept: 4

35. Vertex: (4, 12); y intercept: 4

36. Vertex: (5, 8); y intercept: 2

37. Vertex: (5, 25); additional point on graph: (2, 20)

38. Vertex: (6, 40); additional point on graph: (3, 50)

39. Graph the line y 0.5 x 3. Choose any two distinct

points on this line and find the linear regression model forthe data set consisting of the two points you chose. Exper-

iment with other lines of your choosing. Discuss the rela-

tionship between a linear regression model for two points

and the line that goes through the two points.

40. Graph the parabola y x2 5 x. Choose any three distinct

points on this parabola and find the quadratic regression

model for the data set consisting of the three points you

chose. Experiment with other parabolas of your choice.

Discuss the relationship between a quadratic regression

model for three noncollinear points and the parabola that

goes through the three points.

41. Let f ( x) ( x 1)2

k. Discuss the relationship betweenthe values of k and the number of x intercepts for the

graph of f. Generalize your comments to any function of

the form

f ( x) a( x h)2 k, a 0

42. Let f ( x) ( x 2)2 k. Discuss the relationship be-

tween the values of k and the number of x intercepts for

the graph of f. Generalize your comments to any function

of the form

f ( x) a( x h)2 k, a 0

C

Recall that the standard equation of a circle with radius r and

center (h, k) is

( x h)2 ( y k )2 r 2

In Problems 43–46, use completing the square twice to find the

center and radius of the circle with the given equation.




43. x2 y2 6 x 4 y 36

44. x2 y2 2 x 10 y 55

45. x2 y2 8 x 2 y 8

46. x2 y2 4 x 12 y 24

47. Let f ( x) a( x h)2 k. Compare the values of f (h r )

and f (h r ) for any real number r. Interpret the results in

terms of the graph of f.

48. Let f ( x) ax2 bx c, a 0. Express each of the fol-

lowing in terms of a, b, and c:

(A) The axis of symmetry

(B) The vertex

(C) The maximum or minimum value of f, whichever

exists.

Problems 49–52 are calculus-related. In geometry, a line

that intersects a circle in two distinct points is called a

secant line, as shown in figure (a). In calculus, the line

through the points (x1 , f(x1)) and (x2 , f(x2)) is called a secant line for the graph of the function f, as shown in

figure (b).

In Problems 49 and 50, find the equation of the secant line

through the indicated points on the graph of f. Graph f and the

secant line on the same coordinate system.

49. f ( x) x2 4; (1, 3), (3, 5)

50. f ( x) 9 x2; (2, 5), (4, 7)

51. Let f ( x) x2 3 x 5. If h is a nonzero real number, then

(2, f (2)) and (2 h, f (2 h)) are two distinct points on

the graph of f.

(A) Find the slope of the secant line through these two

points.

(B) Evaluate the slope of the secant line for h 1,

h 0.1, h 0.01, and h 0.001. What value does

the slope seem to be approaching?

Secant line for the graphof a function

(b)

Secant line fora circle

(a)

x

f (x )

(x 1, f (x 1))

(x 2, f (x 2))P

Q

52. Repeat Problem 51 for f ( x) x2 2 x 6.

53. Find the minimum product of two numbers whose diff

ence is 30. Is there a maximum product? Explain.

54. Find the maximum product of two numbers whose sum

60. Is there a minimum product? Explain.

APPLICATIONS

55. Construction. A horse breeder wants to construct a co

next to a horse barn 50 feet long, using all of the barn a

one side of the corral (see the figure). He has 250 feet

fencing available and wants to use all of it.

(A) Express the area A( x) of the corral as a function of

and indicate its domain.

(B) Find the value of x that produces the maximum ar

(C) What are the dimensions of the corral with the

maximum area?

56. Construction. Repeat Problem 55 if the horse breeder

only 140 feet of fencing available for the corral. Does maximum value of the area function still occur at the v

tex? Explain.

57. Projectile Flight. An arrow shot vertically into the air

from a cross bow reaches a maximum height of 484 fe

after 5.5 seconds of flight. Let the quadratic function d

represent the distance above ground (in feet) t seconds

ter the arrow is released. (If air resistance is neglected,

quadratic model provides a good approximation for the

flight of a projectile.)

(A) Find d (t ) and state its domain.

(B) At what times (to two decimal places) will the arr

be 250 feet above the ground?

58. Projectile Flight. Repeat Problem 57 if the arrow reac

a maximum height of 324 feet after 4.5 seconds of flig

59. Engineering. The arch of a bridge is in the shape of a

parabola 14 feet high at the center and 20 feet wide at t

base (see the figure).

x

y

Corral

Horse barn

50 feet




(A) Express the height of the arch h( x) in terms of x and

state its domain.

(B) Can a truck that is 8 feet wide and 12 feet high pass

through the arch?

(C) What is the tallest 8-foot-wide truck that can pass

through the arch?

(D) What (to two decimal places) is the widest 12-foot-

high truck that can pass through the arch?

60. Engineering. The roadbed of one section of a suspension

bridge is hanging from a large cable suspended between

two towers that are 200 feet apart (see the figure). The ca-

ble forms a parabola that is 60 feet above the roadbed at thetowers and 10 feet above the roadbed at the lowest point.

(A) Express the vertical distance d ( x) (in feet) from the

roadbed to the suspension cable in terms of x and state

the domain of d.

(B) The roadbed is supported by seven equally spaced

vertical cables (see the figure). Find the combined

total length of these supporting cables.

61. Break-Even Analysis. Table 1 contains revenue and cost

data for the production of lawn mowers where R is the to-

tal revenue (in dollars) from the sale of x lawn mowers

and C is the total cost (in dollars) of producing x lawn

mowers.

T A B L E 1

x

200

650

1,000

1,350

1,700

R ($)

95,000

275,000

290,000

260,000

140,000

C ($)

145,000

160,000

210,000

230,000

270,000

200 ft

60 ft

x ft

d (x )

x

20 ft

14 fth(x )

(A) Find a quadratic regression model for the revenue

data using x as the independent variable.

(B) Find a linear regression model for the cost data using

x as the independent variable.

(C) Use the regression models from parts A and B to

estimate the x coordinates (to the nearest integer) of

the break-even points.

62. Profit Analysis. Use the regression models computed in

Problem 61 to estimate the indicated quantities.

(A) How many lawn mowers (to the nearest integer) must

be produced and sold to realize a profit of $50,000?

(B) How many lawn mowers (to the nearest integer) must

be produced and sold to realize the maximum profit?

What is the maximum profit (to the nearest dollar)?

63. Water Consumption. Table 2 contains data related to the

water consumption in the United States for selected years

from 1960 to 1990. This data is based on U.S. Geological

Survey, Estimated Use of Water in the United States in1990, circular 1081, and previous quinquennial issues.

(A) Let the independent variable x represent years since

1960. Find a quadratic regression model for the total

daily water consumption.

(B) If daily water consumption continues to follow the

trend exhibited in Table 2, when (to the nearest year)

would the total consumption return to the 1960 level?

64. Water Consumption. Refer to Problem 63.

(A) Let the independent variable x represent years since

1960. Find a quadratic regression model for the daily

water consumption for irrigation.

(B) If daily water consumption continues to follow the

trend exhibited in Table 2, when (to the nearest year)

would the consumption for irrigation return to the

1960 level?

T A B L E 2 Daily Water Consumption

Year

1960

1965

1970

1975

1980

1985

1990

Total (billion gallons)

61

77

87

96

100

92

94

Irrigation (billion gallons)

52

66

73

80

83

74

76

ch02section3

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