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Algebra III Academic
Quadratic Applications
Example #1
When a football is punted it goes up into the air, reaches a maximum altitude, then comes back down. The
number of seconds that have passed since the ball was punted and the height of the ball in feet are related
quadratically.
The equation for our particular problem is: y = -16x2 + 40x + 4
Independent:
Dependent:
a. How many feet above the ground was the ball when it was punted?
b. What was the height of the ball after 4 seconds?
c. When will the ball reach the ground?
d. What was the maximum height of the ball and at what time did this occur?
Example #2
Sam’s local sandwich shop has 3 sandwich sizes, 3’’, 6’’, and 8’’. The cost of these sandwiches are $1.80,
$2.40, and $3.80 respectively. Assume that the cost varies quadratically with its length.
Given the equation y = 10x2 – 70x + 300 (where cost is in cents)
Independent:
Dependent:
a. What is the cost of a 5’’ sandwich?
b. How large is the $6.00 sandwich?
Example #3
Jack Potts dives off the high diving board. His distance in meters from the water varies quadratically with
the number of seconds that have passed since he left the board.
The particular equation is y = -5x2 + 9x + 20
Independent:
Dependent:
a. How high is the diving board?
b. What is Jack’s maximum height above the water?
c.
What is his height after 3.5 seconds? What does this mean?
d.
When does he hit the water?
Space Travel
Phoebe Small is out Sunday driving in her spaceship. As she approaches Mars, she changes her mind,
decides that she does not wish to visit that planet, and fires her retro-rocket. The spaceship slows down, and
if all goes well, stops for an instant then starts pulling away. While the rocket’s motor is firing, Phoebe’s
distance in kilometers from the surface of Mars depends on the number of minutes since she started firing
the rocket.
The equation is y = 3x2 – 78x + 500
Independent:
Dependent:
a. Find the distance intercept and tell what this number represents in this problem.
b. According to the equation, where will Phoebe be when t = 15? When t = 16? Is she pulling away from
Mars when t = 16 or is she still approaching?
c.
Did Phoebe end up landing on Mars before pulling away? Support your answer mathematically!
Flying Rock
Chuck Stone is standing atop a high platform. He fires a rock up into the air with his slingshot. While it is
in flight, the rock’s distance above the ground (meters) is a quadratic function of time (seconds).
The equation is y = -5x2 + 43x + 30
Independent:
Dependent:
a. What is the highest the rock will be above the ground?
b. How high is the platform?
c. Where is the rock 4 seconds after being fired?
d.
When will the rock hit the ground?
e.
What if the rock fell into a well and splashed into the water 10 seconds after Chuck fires it. How deep
is the well?
Actuary
Suppose you are an actuary for an Insurance Agency. Your company plans to offer a senior citizen’s
accident policy, and you must predict the likelihood of an accident as a function of the driver’s age.
The equation is y = 0.4x2 – 36x + 1000
Note: Age is in years and the accidents are per 100 million km driven.
Independent:
Dependent:
a. How many accidents would you expect for an 80 year old driver?
b. Based on our model, who is safer, a 16 year old driver or a 70 year old driver?
c. What age driver appears to be the safest?
d.
What is the domain if the company decides to insure drivers up to 830 accidents per million km?
Rectangular Walkway
A rectangular pool 5 meters by 7 meters is to be surrounded by a walkway of width x meters. This problem
concerns the rectangular region taken up by the pool and walkway.
a. Draw a picture of the pool and walkway.
b. Find the length and width of the rectangle.
c. Write the area of the region as a function of x.
d. Predict the area of the region if x = 1, 2, and 3.
e. Find the value of x which makes the region have an area of 150 square meters.
Homework: Selected Problems from the text or worksheet…tba.