Warm up!
Write an equation in vertex form of the quadratic with
vertex (1, 3) and passed through the point (2, 0)
Day 38 Topic
Quadratics
Objective
The student will be able to solve quadratic word problems.
Modeling Dropped Objects
When and object is dropped, its height h (in feet) above the ground after t seconds can
be modeled by the function:
ℎ = −16𝑡 2 + ℎ0
where h0 is the object’s initial height (in feet)
The graph of ℎ = −16𝑡 2 + 200 is modeled below.
What is the initial height that the object is being dropped from?
Ex.1
For a science competition, students must design a container to prevent an egg from
breaking when dropped from a height of 50 feet. How long des the container take to hit
the ground?
Recall: ℎ = −16𝑡 2 + ℎ0
Ex.2
The tallest building in the United States is in Chicago, Illinois? It is 1450 feet tall.
a) How long would it take a penny to drop from the top of this building?
b) How fast would the penny be traveling when it hits the ground if the speed is given by
𝑠 = 32𝑡, where t is the number of seconds since the penny was dropped?
Ex.3
How long would it take an object dropped from a 550-foot tall tower to land on
the roof of a 233-foot tall building?
Ex.4
A coyote is standing on a cliff 254 feet above the roadrunner. If the coyote drops
a boulder from the cliff, how much time does the roadrunner have to move out
of its way?
Object is DROPPED: ℎ = −16𝑡 2 + ℎ0
Object is THROWN: ℎ = −16𝑡 2 + 𝑣𝑡 + ℎ0
h = height (feet)
h0 = initial height (feet)
t = time in motion (seconds)
v = initial velocity (feet per second)
Ex.5
A child throws a ball upward with an initial upward velocity of 15 ft/s from a
height of 2 feet. If no one catches the ball, after how many seconds will it land?
ℎ = −16𝑡 2 + ℎ0
or
ℎ = −16𝑡 2 + 𝑣𝑡 + ℎ0
Ex.6
A football is kicked from a starting height of 3 ft. with an initial upward velocity of
40 ft/s. Will the football ever reach a height of 30ft.?
ℎ = −16𝑡 2 + ℎ0
or
ℎ = −16𝑡 2 + 𝑣𝑡 + ℎ0
Ex.7
A juggler tosses a ball into the air. The ball leaves the juggler’s hand 4 feet
above the ground and has an initial velocity of 40 feet per second. The juggler
catches the ball when it falls back to a height of 3 feet. How long is the ball in
the air?
ℎ = −16𝑡 2 + ℎ0
or
ℎ = −16𝑡 2 + 𝑣𝑡 + ℎ0
Ex.8
Solve for x.
Area = 24.5 in2
x + 4.2
x
Ex.9
For the years 1989-1996, the amount A (in billions of dollars) spend on long
distance telephone calls in the US can be modeled by 𝐴 = 0.560𝑡 2 + 4.88𝑡 + 51
where t is the number of years since 1989. In what year did the amount spent
reach $60 billion?
1. You jump off a 15-foot diving board with an initial upward velocity of 3 ft/sec. How long will
it take you to hit the water?
2. You drop your keys from a window 30 feet above ground to your friend below. Your friend
doesn’t catch them so they hit the ground. How long will it take for your keys to hit the
ground?
3. A lacrosse player throws a ball upward from her playing stick with an initial height of 7
feet, at an initial velocity of 90 feet per second. How long will it take the ball to hit the
ground?
4. A falcon dives toward a pigeon on the ground. When the falcon is at a height of 100 feet
the pigeon sees the falcon, which is diving at 220 feet per second. Estimate the time the
pigeon has to escape.
5. An acorn falls 45 feet from the top of a tree. How long will the acorn take to hit the
ground?
6.
You are standing on a bridge over a creek, holding a stone 20 feet above the water. You
release the stone. How long will it take the stone to hit the water?