Applications of Quadratic Functions
Note: The height of a launched object or a falling object can be modeled by the quadratic function
ℎ(𝑡) = −16𝑡 2 + 𝑣0 𝑡 + 𝑠0 .
In this equation:
𝑡 represents _______________________________ and is measured in ________________.
ℎ(𝑡) represents _______________________________ and is measured in ________________.
𝑣0 represents _______________________________ and is measured in ________________.
𝑠0 represents _______________________________ and is measured in ________________.
1. Orioles centerfielder Adam Jones throws a baseball from a height of 5 feet at an upward velocity of 30
feet per second.
a. How long will it take the baseball to hit the ground?
b. What is the maximum height of the baseball?
c. Find the average rate of change on [0, .5] and interpret it in the context of the problem.
d. If he is standing 358 feet from home plate, how much would he need to increase his
upward velocity in order to get it there without bouncing?
2. Because the Redskins can’t score touchdowns, they are trying to score a field goal. If their kicker
kicks the ball from the ground with an upward velocity of 36 feet per second…
a. How long will it take for the ball to reach the field goal?
b. How long will it take the ball to hit the ground?
c. What is the maximum height of the ball?
d. How could the upward velocity be adjusted so that the maximum height is increased by10
feet?
3. A diver jumps from a platform 10 meters above the surface of the water. The diver’s height is given
by the equation ℎ(𝑡) = −4.9𝑡 2 + 3.5𝑡 + 10, where 𝑡 is the time, in seconds, after the diver jumps.
a. For what time 𝑡 is the diver 1 meter above the water?
b. What is the maximum height of the diver?
c. What is the platform were only 5 feet high? What would the maximum height be then?
d. What is the diver’s average rate of change over the first half second of the drive?
4. The height, in meters, of a model rocket on a particular launch can be modeled by the equation ℎ(𝑡) =
−4.9𝑡 2 + 102𝑡 + 100, where 𝑡 is the time, in seconds, after its engine burns out 100 meters above the
ground.
a. When will the rocket reach a height of 600 meters?
b. When will it reach a height of 450 meters during its descent?
c. When will it hit the ground?
d. If the engine burns out 20 feet sooner than planned, but all other parameters stay the same,
how high will the rocket reach?
5. The Red Sox are playing the Tampa Bay Rays at Tropicana field, which has a dome that is 175 feet
high. Xander Bogaerts steps up to the plate and gets a hit. Its height as a function of time can be
modeled by the function 𝑓(𝑥) = −16𝑥 2 + 100𝑥 + 3.5.
a. Will it hit the roof?
b. If so, how long will it take to do so?
c. How long will the ball take to land?
d. If Xander really wanted to hit the roof, what would he have to change in the function?
e. What would be an example of a function for which the ball would hit the roof?
6. You have just purchased a Civil War era cannon from a shop in Gettysburg and are trying it out in
the FHS gym. (Sh, don’t tell Principal Campagnioli). The gym roof is 42 feet high and the cannon sits
only 1.5 feet off the ground.
a. Guess the minimum initial upward velocity required to put a hole in the ceiling?
b. What about to miss the ceiling by a few feet?
c. What about to just barely touch the ceiling?
d. Find the length of time the cannonball will be in the air if it just grazes the ceiling.