Name
Oct. 23, 2015
Math 4 problem set
polynomial application problems page 1
Objective: Apply polynomial functions to solve problems about real-world situations.
Generally, the first step in these application problems is to write a function that fits the problem
situation (unless the function was already given to you in the problem). From there, there are a
few different possibilities:
mathematical task
Finding the y value(s) for a
specified x value.
Finding the x value(s) for a
specified y value.
Finding zeros (x-intercepts).
Finding a maximum or
minimum.
how to do it
(by hand or by calculator)
Input the x value into the
function formula and evaluate.
Solve an equation algebraically,
or graph a horizontal line at that
y value then use 2nd Calc
Intersect.
Solve an equation algebraically,
or use 2nd Calc Zero.
For quadratics: Use any of the
methods we’ve studied for
finding the vertex.
For any function: Use 2nd Calc
Maximum or Minimum.
typical application
(for a moving object)
How high is the object at a
specified time?
At what time(s) is the object at
a specified height?
When does the object reach
the ground?
What is the maximum height
of the object, and when does it
reach this height?
1. A model rocket is launched from a platform that’s 8 feet above the ground. The height of the
rocket, measured in feet, t seconds after the launch, is given by
f(t) = –16t2 + 70t + ____
(If you need help: see page 167 for parts a/g/h, and the chart above for the other parts.)
a. Fill in the ____ term of the function formula.
b. Evaluate f(3) and explain the meaning of the answer as it relates to the rocket.
c. Does the object reach a height of 80 feet? If so, at what time(s)?
d. Find the vertex of f(t) using two different methods, and confirm that the two answers
agree with each other.
e. What is the rocket’s maximum height, and at what time is it reached?
f. After how much time does the rocket hit the ground?
g. What was the initial vertical velocity of the rocket?
h. Bonus: How fast was the rocket going when it hit the ground? (see page 167 for the
velocity formula)
Name
Oct. 23, 2015
Math 4 problem set
polynomial application problems page 2
2. The publisher of a magazine that currently sells 80,000 copies per issue for $1.60 each
decides to raise the magazine’s price. By surveying readers of the magazine, the publisher
has found that the magazine will lose 2,000 readers for every $0.10 increase in price.
This problem will walk you through finding the answer to the following question:
If the publisher’s goal is to maximize revenue, what selling price should she choose?
Let x stand for the number of $0.10 increases in price. (For example, x = 2 would mean that
the price increases from $1.60 to $1.80.)
a. Write a formula for the magazine’s selling price as a function of x.
b. Write a formula for the number of copies sold as a function of x.
c. Write a formula for the total revenue as a function of x.
d. What x value would maximize the revenue?
e. What selling price should the publisher choose?
3. A rectangular dog pen is to be constructed using a barn wall for one side and a total of
60 yards of fencing for the other three sides, as illustrated below.
a. Let x stand for the length marked in the diagram. Label the other side lengths of the dog
pen in terms of x.
b. If x = 10 yards, what are the other dimensions of the dog pen and the area of the dog pen?
c. Write a function formula for the area of the dog pen.
d. Find the dimensions of the dog pen that would maximize the area.
Name
Oct. 23, 2015
Math 4 problem set
polynomial application problems page 3
4. A manufacturer makes rectangular cardboard boxes through the following process: beginning
with a 15"-by-60" sheet of cardboard, four congruent squares are removed from the corners,
then four folds are made to form an open-top box, as illustrated in this diagram.
a. Just for part a of this problem, suppose that the side length of the cut-out squares is 3".
That is, x = 3 in the above diagram. Find the three lengths of the cardboard box,
then calculate the box’s volume.
b. Now let x stand for the side length of the squares, and let f(x) stand for the box’s volume.
Write a function formula for f(x).
c. Explain why it would be impossible to have x = 10.
d. What is the domain of function f(x)? In other words, what is the interval of possible
x values that make sense in the problem situation?
e. Is it possible using the above process to produce a box whose volume is 1000 cubic
inches? If yes, find the possible dimensions of the box. If no, explain why not.
f. What x value will produce the box of greatest possible volume, and what will be the
volume of that box?
5. An orange grower has 400 crates of fruit ready for market and will have 20 more for each
day the grower waits. The present price is $240 per crate and will drop an estimated $8 for
each day waited. To maximize income, in how many days should the grower ship the crop?
Credits: Some problems adapted from Advanced Mathematics by Richard Brown.