Mathematics - Algebra - Functions

Exercise 2.5
15.
Cell Phones The following table gives the number of millions of U.S. cellular telephone subscribers.
a. Create a scatter plot for the data with x equal to the number of years from 1985. Does it appear that the data could be modeled with a quadratic function?
b. Find the quadratic function that is the best fit for these data, with x equal to the number of years from 1985 and y equal to the number of subscribers in millions?
c. Use the model to estimate the number in 2005.
d. What part of the U.S. population does this estimate equal?

Year Subscribers(millions) Year Subscribers(millions)
1985 0.340 1994 24.134
1986 0.682 1995 33.786
1987 1.231 1996 44.043
1988 2.069 1997 55.312
1989 3.509 1998 69.209
1990 5.283 1999 86.047
1991 7.557 2000 107.478
1992 11.033 2001 128.375
1993 16.009 2002 140.767

25.
World Population One projection of the world population by the United Nation for selected years (a low projection scenario) is given in the table below.

Year Projected Population(million) Year Projected Population(million)
1995 5666 2075 6402
2000 6028 2100 5153
2025 7275 2125 4074
2050 7343 2150 3236
a. Find a quadratic function that fits these data, using the number of the years after 1990 as the input.
b. Find the positive x-intercept of this graph, to the nearest year.
c. When can we be certain that this model no longer applies?

31.
Classroom Size The date in the table below give the number of students per teacher for selected years between 1960 and 1998.

Year Students per Teacher Year Students per Teacher
1960 25.8 1992 17.4
1965 24.7 1993 17.4
1970 22.3 1994 17.3
1975 20.4 1995 17.3
1980 18.7 1996 17.1
1985 17.9 1997 17.0
1990 17.2 1998 17.2
1995 17.3

a. Find the power function that is the best fit for the data, using as input the number of years after 1950.
b. According to the unrounded model, how many students per teacher were there in 2000?
c. Is this function increasing or decreasing during this time period?
d. What does the model predict will happen to the number of student per teacher as time goes on?

35. Insurance Rates The following table gives the monthly insurance rates for a $100,000 life insurance policy for smokers 35-50 years of age.
a. Create a scatter plot for the data.
b. Does it appear that a quadratic function can be used to model the data? If so, find the best-
fitting quadratic model?
c. Find the power model that is the best fit for the data.
d. Compare the two models by graphing each model on the same axes with the data points.
Which model appears to be better fit?

Age(yr) Monthly Insurance Rate ($) Age(yr) Monthly Insurance Rate ($)
35 17.32 43 23.71
36 17.67 44 25.11
37 18.02 45 26.60
38 18.46 46 28.00
39 19.07 47 29.40
40 19.95 48 30.80
41 21.00 49 32.55
42 22.22 50 34.47

Exercise 2.6
41.
Population of Children The following table gives the estimate population (in millions) of U.S. boys age 5 and under and the estimate U.S. population (in millions) of girls age 5 and under in selected years.
Year 1995 2000 2005 2010
Boys 10.02 9.71 9.79 10.24
Girls 9.57 9.27 9.43 9.77

A function that models the population (in millions) of U.S. boys age 5 and under t years after 1990 is B(t) =0.0076t²-0.1752t+10.705, and a function that model the population (in millions) of U.S. girls age 5 and under t years after 1990 is G(t) =0.0064t² - 0.1448t+10.12.
a. Find the equation of a function that models the estimate U.S population (in millions) of children age 5 and under t years after 1990.
b. Use the result of part (a) to estimate the U.S population of children age 5 and under in 2003.

Exercise 2.7
51.
Education If the function f(x) gives the number of female PhDs produced by American universities x years after 1990 and the function g(x) gives the number of male PhDs produced by American universities x years after 1990, what function gives the total number of PhDs produced by American universities x years after 1990?

45.
Projectiles Two projectiles are fired into the air over a lake, with the height of the first projectile given by y=100+130t-16t² and the height of the second projectile given by
y=-16t²+180t, where y is in feet and t is in seconds. Over what time interval, before the lower one hits the lake, is the second projectile above the first?