Algebra - Solving Polynomial & Rational Functions

Exercise 4.5
27. Advertising and Sale The monthly sales volume y (in thousand of dollars) of a product is related to monthly advertising expenditures x (in thousands of dollars) according to the equation.
Y= 400x
x+20
a. What monthly sales will result if $5000 is spent monthly on advertising?
b. What value of x makes the denominator of this function 0? Will this ever happen in the context of this problem?

29. Average Cost The average cost per set for the production of television sets is given by

_ 1000+30x+0.1x²
C(x) = x
where x is the number of thousands of units produced.
a. Graph this function using the window [-10, 10] by [-200,300].
b. Graph this function using the window [0, 50] by [0,300]
c. Which window makes sense for this application?
d. Use the graph with the window [0,200] by [0,300] to find the minimum average cost, and the number of units that gives the minimum average cost.

33. Cost-Benefit Suppose the cost C of removing p% the impurities from the waste water in a manufacturing process is given by
3600p
C (p) = 100-p
a. Where does the graph of this function have a vertical asymptote?
b. What does this tell us about removing the impurities from this process?

35. Sales Volume Suppose the weekly sales volume (in thousands of units) for a product is given by

640
V = (p + 2)²
where p is the price in dollars per unit.
a. Graph this function on the p-interval [-10, 10]. Does the graph of this function have a vertical asymptote on this interval? Where?
b. Complete the table below.
c. What values of p gives a weekly sales volume that makes sense? Does the graph of this function have a vertical asymptote on this interval?
d. What is the horizontal asymptote of the graph of the weekly sales volume? Explain what this means.
Price?Unit($) 5 20 50 100 200 500
Weekly Sales Volume

43. Farm Workers The percents of U.S. workers in the farm occupation during certain years from 1820 are show in the table below.

Year Percent of workers who are farm workers Year Percent of workers who are farm workers
1820 71.8 1950 11.6
1850 63.7 1960 6.1
1870 53 1970 3.6
1900 37.5 1980 2.7
1920 27 1985 2.8
1930 21.2 1990 2.4
1940 17.4 1994 2.5

Assume that the percent can be modeled with the function

R(t) = -8091.2t + 1,558,900
1.09816t² - 122.183 + 21,472.6

where t is the number of years after 1800.
a. Graph this function and the data point from the table on the same axes, using the window [0,250] by [0, 80].
b. Compare the percent of farm works in 1930 from the table of value and the functional model.

Exercise 4.6
27. Advertising and Sales The monthly sales volumes y (in thousands of dollars) is related to monthly advertising expenditure x (in thousands of dollars) according to the equation.

400x
y = x + 20
Spending how much money on advertising will result in sales of at least $200,000 per month?

29. Future Value The future value of $2000 invested for 3 years at rate r, compounded annually, is given by S=2000(1+r)³. Find the rate r that gives a future value from $2662 to $3456, inclusive.

35. Population Suppose the number of employees of a start-up company is given by
30 + 40t
F (t) = 5 + 2t
Where t is the number of months after the company is organized.
a. For what values of t is f (t) < 18?
b. During what months is the number of employees below 18?