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Project I – Quadratic Functions Project
One of the many applications of quadratic functions in called the Profit Parabola. The Profit Parabola can be seen if we investigate the following scenario:
The business manager of a
90
unit apartment building is trying to determine the rent to be charged. From past e
x
perience with similar buildings, when rent is set at
$400
, all the units are full. For every $20 increase in rent, one additional unit remains vacant. What rent should be charged for maximum total revenue? What is that maximum total revenue?
1
. To help solve the above scenario, perform an internet search for Profit Parabola or Applications of Quadratic Functions. List the URL of one of the applications that you find.
URL
http://www.mathissfun.com/algebra/quadratic-equation.html____________________________________________________________
2. Go to
PurpleMath “Max/Min” Word Problems
to see the process used for determining the quadratic function for revenues R(x) as a function of price hikes x on page
3
with the canoe-rental business problem. Use this process to determine the quadratic function that models the revenues R(x) as a function of price hikes x in the apartment building scenario above. SHOW ALL YOUR WORK!
|
Rent hikes |
Rent per apartment |
Number of rentals |
Total revenue |
| $400 | 90 |
400(90) =36,000 |
|
| 1 |
$420 |
90-(1) |
420(89) =37,380 |
| 2 |
$440 |
90-(2) |
440(88) =38,720 |
| 3 |
$460 |
90-(3) |
460(87) =40,020 |
| x |
400+x (20) |
400+x (20) (90-x) =0 |
3. What is the formula for revenues R after x $20 price hikes in the apartment building?
R= (90-x)(400+20x)
4. Graph the function. Clearly label the graph (desmos.com is a great an on-line graphing resource).
5. Find the maximum revenue (or income) of the apartment building.
6. What is the rent that coincides with this maximum revenue?
7. What is the outcome of the rent hike of $20 results in 2 additional vacancies instead of 1 additional vacancy? Recalculate questions 3, 5, 6 for this new scenario.
8. Set up a similar scenario, of your own invention, using a business that you are interested in. Write up the scenario (problem) and the solution process involved. Find the solution to the problem you invented. Graph the function and attach it.
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