1.9
Function Models
and Applications
Practice Set 4
Problems 1 − 3, express a rational function that describes the scenario.
1. A large mixing tank contains 250 gallons of a solution, into which 15 pounds of salt have been
mixed. A faucet will be opened to allow additional solution to be pumped into the tank at a rate
of 15 gallons per minute. At the same time salt is poured into the tank at a rate of 3 pounds per
minute. Write an equation 𝐶𝐶(𝑡𝑡) that represents the concentration (pounds per gallon) of salt in
the tank after 𝑡𝑡 minutes.
2. A rectangular box with a square base will have a volume of 20 cubic feet. The material for the top
and the base costs $.25 per square foot. The material for the sides cost $.10 per square foot. Let
𝑥𝑥 = the length of the side of the base. Write an equation 𝐶𝐶(𝑥𝑥) that would represent the cost of the
box.
3. An oil storage tank in the shape of a right circular cylinder has a volume of 55 gallons. Write an
equation in terms of the radius, 𝑟𝑟, that represents the surface area 𝐴𝐴(𝑟𝑟). Note that the tank will
have both a top and bottom to seal the container when filled with oil.
Problems 4-11, answer each question.
4. A power plant burns coal for generating electricity. The cost of removing 𝑝𝑝𝑝 of pollutants from
52000𝑝𝑝
the smokestack emissions, 𝐶𝐶(𝑝𝑝) is given by 𝐶𝐶 𝑝𝑝 = 100−𝑝𝑝 . The plant currently removes 70% of
the pollutants. A new law will require 85% of the pollutants must be removed. How much more
will the power plant spend to meet the new requirements?
© 2022 Jean Adams
Flamingo Math.com
5. The concentration C , in milligrams per liter, of an antibiotic in a patient’s bloodstream t hours
60𝑡𝑡
after injection is given by 𝐶𝐶 𝑡𝑡 = 𝑡𝑡 2 +25.
A. What happens to the concentration of the drug as t increases? Think about end behavior.
B. Sketch the function from 𝑡𝑡 = 0 to 𝑡𝑡 = 20 in the
grid provided at right.
C. Use your calculator to determine the time at
which the concentration is highest.
6. The daily cost C of manufacturing electric scooters is given by 𝐶𝐶 𝑥𝑥 = 95𝑥𝑥 + 5400. The average
95𝑥𝑥+5400
daily cost 𝐶𝐶̅ is given by 𝐶𝐶̅ 𝑥𝑥 =
. How many scooters must be produced each day for the
𝑥𝑥
average daily cost to be no more than $125?
7. Carry-on luggage for an airline has a length 10 inches greater than the depth. The sum of the
length, width, and depth may not exceed 45 inches. Let 𝑥𝑥 = depth of the carry-on.
A. Write an equation relating volume to depth x.
B. Graph the function and find the x-intercepts. State a reasonable domain.
C. Find the maximum possible volume, to the nearest cubic inch and the corresponding
dimensions of the carry-on bag.
© 2022 Jean Adams
Flamingo Math.com
8. The number of elk P at any time t (in years) in a national game reserve is given by P 𝑡𝑡 =
A. Find the number of elk when t is 25, 42, 100.
600+560𝑡𝑡
.
20+0.8𝑡𝑡
B. Find the horizontal asymptote of the graph 𝑦𝑦 = 𝑃𝑃(𝑡𝑡)
C. According to the model, what is the largest possible elk population?
9. Home Run Sports Apparel is selling baseball caps for $12.50 each. It costs $3.25 to produce each
cap, and the weekly overhead costs is $4500.
A. Let x be the number of caps produced each week. Express the average cost (including
overhead costs) of production one cap as a function of x.
B. Solve algebraically to find the number of baseball caps that must be sold each week to make
a profit. Check your answer graphically.
C. How many caps must be sold to make a profit of $1000 in one week. Support your answer
algebraically.
© 2022 Jean Adams
Flamingo Math.com
10. Consider all rectangles with an area of 220 ft2. Let x be the length of one side of such a rectangle.
A. Express the perimeter P as a function of x.
B. Find the dimensions of the rectangle that has the least perimeter.
C. What is the least perimeter?
11. The cost, in dollars, of processing x pounds of sugarcane each day can be modeled by the cost
function 𝐶𝐶 𝑥𝑥 = 0.001𝑥𝑥 3 − 0.12𝑥𝑥 2 + 6𝑥𝑥 + 250. Economists define the average cost function as
A. Write the average cost function.
𝐶𝐶 𝑥𝑥 =
𝐶𝐶 (𝑥𝑥)
𝑥𝑥
B. What is the average cost of processing 100 pounds of sugarcane?
C. Use your graphing utility to graph the average
cost function on grid at right.
D. Find the number of pounds of sugarcane that should be
processed to minimize the average cost per day.
E. What is the minimum average cost?
© 2022 Jean Adams
Flamingo Math.com