ANALISIS DE RESULTADOS FINANCIEROS Y OPERATIVOS AL 30 DE JUNIO 2005

EXERCISES

1. A homing pigeon starts 1000 miles from home and flies 50 miles toward home each day. Express distance from home in miles,D, as a function of number of days, d.

2. You buy a saguaro cactus 5 ft high and it grows at a rate of 0.2 inches each year. Express its height in inches,h, as a function of timet in years since the purchase.

3. The temperature of the soil is 30◦_{C at the surface and}

decreases by0.04◦_{C for each centimeter below the sur-}

face. Express temperatureT as a function of depth d, in centimeters, below the surface.

Give the values forb and m for the linear functions in Exer- cises4–9.

4. f (x) = 3x + 12 5. g(t) = 250t − 5300 6. h(n) = 0.01n + 100 7. v(z) = 30

8. w(c) = 0.5c

9. u(k) = 0.007 − 0.003k

10. The cost,$C, of hiring a repairman for h hours is given byC = 50 + 25h.

(a) What does the repairman charge to walk in the door?

(b) What is his hourly rate?

11. The cost,$C, of renting a limousine for h hours above the 4 hour minimum is given byC = 300 + 100h. (a) What does the300 represent?

(b) What is the hourly rate?

12. The population of a town,t years after it is founded, is given byP (t) = 5000 + 350t.

(a) What is the population when it is founded? (b) What is the population of the town one year after it

is founded? How much does it increase by during the first year? During the second year?

In Exercises13–23, identify the initial value and the rate of change, and explain their meanings in practical terms.

13. An orbiting spaceship releases a probe that travels di- rectly away from Earth. The probe’s distances (in km) from Earth aftert seconds is given by s = 600 + 5t. 14. After a rain storm, the water in a trough begins to evap-

orate. The amount in gallons remaining aftert days is given byV = 50 − 1.2t.

15. The monthly charge of a cell phone is25 + 0.06n dol- lars, wheren is the number of minutes used.

16. The number of people enrolled in Mathematics101 is 200 − 5y, where y is the number of years since 2004. 17. On a spring day the temperature in degrees Fahrenheit

is50 + 1.2h, where h is the number of hours since noon.

18. The value of an antique is2500 + 80n dollars, where n is the number of years since the antique is purchased. 19. A professor calculates a homework grade of100 − 3n

forn missing homework assignments.

20. The cost,C, in dollars, of a high school dance attended byn students is given by C = 500 + 20n.

21. The total amount,C, in dollars, spent by a company on a piece of heavy machinery aftert years in service is given byC = 20,000 + 1500t.

22. The population,P , of a city is predicted to be P = 9000 + 500t in t years from now.

23. The distance,d, in meters from the shore, of a surfer riding a wave is given byd = 120 − 5t, where t is the number of seconds since she caught the wave. In Exercises24–29, identify the slope andy-intercept and graph the function.

24. f (x) = 2x + 3 25. f (x) = 4 − x 26. f (x) = −2 + 0.5x 27. f (x) = 3x − 2 28. f (x) = −2x + 5 29. f (x) = −0.5x − 0.2

PROBLEMS

30. If the tickets for a concert costp dollars each, the num- ber of people who will attend is2500 − 80p. Which of the following best describes the meaning of the80 in this expression?

(i) The price of an individual ticket.

(ii) The slope of the graph of attendance against ticket price.

(iii) The price at which no one will go to the concert. (iv) The number of people who will decide not to go if

5.2 WORKING WITH LINEAR EXPRESSIONS 121

31. Long Island Power Authority charges its residential customers a monthly service charge plus an energy charge based on the amount of electricity used.2 _The

monthly cost of electricity is approximated by the func- tion:C = f (h) = 36.60 + 0.14h, where h represents the number of kilowatt hours (kWh) of electricity used in excess of250 kWh.

(a) What does the coefficient0.14 mean in terms of the cost of electricity?

(b) Findf (50) and interpret its meaning.

32. The following functions describe four different collec- tions of baseball cards. The collections begin with dif- ferent numbers of cards and cards are bought and sold at different rates. The number,B, of cards in each col- lection is a function of the number of years,t, that the collection has been held. Describe each of these collec- tions in words.

(a) B = 200 + 100t (b) B = 100 + 200t (c) B = 2000 − 100t (d) B = 100 − 200t

33. Match the functions in (a)–(e) with the graphs in Fig- ure 5.8. The constants s and k are the same in each function. (a) f (x) = s (b) f (x) = kx (c) f (x) = kx − s (d) f (x) = 2s − kx (e) f (x) = 2s − 2kx (iv) (v) (iii) (i) (ii) x y Figure 5.8

34. The velocity of an object tossed up in the air is modeled by the functionv(t) = 48 − 32t, where t is measured in seconds, andv(t) is measured in feet per second. (a) Create a table of values for the function. (b) Graph the function.

(c) Explain what the constants 48 and −32 tell you about the velocity.

(d) What does a positive velocity indicate? A negative velocity?

35. Ifa is a constant, does the equation y = ax +5a define y as a linear function of x? If so, identify the slope and vertical intercept.

36. The graphs of two linear functions have the same slope, but differentx-intercepts. Can they have the same y- intercept?

37. The graphs of two linear functions have the samex- intercept, but different slopes. Can they have the same y-intercept?

Give the slope andy-intercept for the graphs of the functions in Problems38–43. 38. f (x) = 220 − 12x 39. f (x) = 1 3x − 11 40. f (x) = x 7− 12 41. f (x) = 20 − 2x 3 42. _{f (x) = 15 − 2(3− 2x)} 43. f (x) = πx

44. Ifn birds eating continuously consume V in3

of seed in T hours, how much does one bird consume per hour?

45. Ifn birds eating continuously consume W ounces of seed inT hours, what are the units of W/(nT )? What doesW/(nT ) represent in practical terms?

5.2

WORKING WITH LINEAR EXPRESSIONS

An expression, such as 3 + 2t, that defines a linear function is called a linear expression. When we are talking about the expression, rather than the function it defines, we call b the constant term and mthe coefficient.

Example 1 Identify the constant term and the coefficient in the expression for the following linear functions. (a) u(t) = 20 + 4t _(b) v(t) = 8 − 0.3t (c) w(t) = t/7 + 5

122 Chapter 5 LINEAR FUNCTIONS, EXPRESSIONS, AND EQUATIONS

Solution (a) We have constant term 20 and coefficient 4.

(b) Writing v(t) = 8 + (−0.3)t, we see that the constant term is 8 and the coefficient is −0.3. (c) Writing w(t) = 5 + (1/7)t, we see that the constant term is 5 and the coefficient is 1/7.

Different forms for linear expressions reveal different aspects of the functions they define.