Math IV Name______

Applications of Rational Functions

  1. A utility company burns coal to generate electricity. The cost of removing a certain percent of the pollutants form the smokestack emissions is typically not a linear function. That is, if it costs C dollars to remove 25% of the pollutants, it would cost more than 2C dollars to remove 50% of the pollutants. As the percent of removed pollutants approaches 100%, the cost tends to become prohibitive. Suppose that the cost C (in dollars) of removing p% of the smokestack pollutants is

C = 80,000p0 ≤ p ≤ 100.

100 – p

Suppose you are a member of a state legislature that is considering a law that would require utility companies to remove 90% of the pollutants from their smokestack emissions. If the current law required 85% removal, how much additional cost would there be to the utility company because of the new law?

  1. In a pilot project, a rural township was given recycling bins for separating and storing recyclable products. The cost (in dollars) for supplying bins to p% of the population is

C = 25,000p0 ≤ p ≤ 100.

100 - p

  1. Find the cost of giving bins to 15% of the population.
  2. Find the cost of giving bins to 50% of the population.
  3. Use a graphing utility to graph the cost function. What was your viewing window? Why did you choose the values you did for the window?
  4. According to this model, would it be possible to supply bins to 100% of the residents? Explain.
  1. The game commission introduces 100 deer into newly acquired state game lands. The population of the herd is

N = 20(5 + 3t)t ≥ 0 where t is the time (in years).

1 + 0.04t

  1. Find the population when t is 5, 10, and 25.
  2. What is the limiting size of the herd as time increases? Explain your reasoning.
  1. The number of United States military reserve personnel M (in thousands) for the years 1990 through 1997 can be modeled by

Y = 1671.92 + 130.23twhere t is time (in years), with t=0

1 – 0.02t + 0.02t2corresponding to the year 1990.

a. Use the model to estimate the number of military reserve personnel in

2012. In 2020.

b. Would this model be useful for estimating the number of military

personnel for future years? Explain.

  1. The concentration of a certain chemical in the bloodstream t hours after injection into muscle tissue is

C = 3t2 + t t ≥ 0.

t3 + 50

  1. Determine the horizontal asymptote of the function and interpret its meaning in the context of the problem
  2. Use a graphing utility to graph the function and approximate the time when the bloodstream concentration is greatest.
  3. Use a graphing utility to determine when the concentration is less than 0.345.
  1. A rectangular page is designed to contain 48 square inches of print. The margins on each side of the page are each 1 ½ inches. The margins at the top and bottom are each 1 inch. What should the dimensions of the page be so that the minimum amount of paper is used?
  1. Let A be the area to be minimized. From the figure, write an expression for area using A = length x width.
  2. The printed area inside the margins is modeled by length x width = 48, or xy = 48. Solve this equation for y.
  3. Rewrite the equation for A in terms of just one variable by substituting your answer in part b for y.
  4. Simplify the expression.
  5. Graph the rational function.
  6. State the domain of the function.
  7. Find the relative minimum.
  8. State the dimensions of the page.