1. By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org
Quadratic Applications Vertical Motion & Profit / Income
By: Jeffrey Bivin
Lake Zurich High School
Last Updated: November 30, 2007

2. Vertical Motion
Compares the height of an object with the time in flight.
g = force of gravity: ft/sec or 9.8 m/sec
vo = initial velocity
ho = initial height
Jeff Bivin -- LZHS

3. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec.
What is the maximum height of the ball?
When will the ball reach the maximum height?
When will the ball return to the ground?
When will the ball be at a height of 250 feet?
When will the ball be at a height of 400 feet?
When will the ball be at a height of 50 feet?
If the ball lands in a 20 foot deep pit, when will the ball hit the bottom of the pit?
What will be the height of the ball in 3 seconds?
How far from the building will the ball land?
Jeff Bivin -- LZHS

4. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec.
What is the maximum height of the ball?
We need to use:
g = 32 ft/s2
vo = 80 ft/s
ho = 200 ft ?
200 ft
Jeff Bivin -- LZHS

5. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec.
What is the maximum height of the ball?
300 ft.
Where is the maximum?
Find the vertex……
Vertex is:
Jeff Bivin -- LZHS

6. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec.
When will the ball reach the maximum height?
2.5 sec
Where is the maximum?
Find the vertex……
Vertex is:
Jeff Bivin -- LZHS

7. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec.
When will the ball return to the ground?
6.830 sec.
What is the height at the ground?
h(t) = 0
200 ft
Get the decimal approximations:
Jeff Bivin -- LZHS

8. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec.
When will the ball be at a height of 250 feet?
0.732 sec & sec.
250 feet
What height?
h(t) = 250 ?
200 ft
Get the decimal approximations:
Jeff Bivin -- LZHS

9. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec.
When will the ball be at a height of 400 feet?
never
What height?
h(t) = 400
so?
Wait, what was the maximum height?
Jeff Bivin -- LZHS

10. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec.
When will the ball be at a height of 50 feet?
6.453 sec.
What height?
h(t) = 50
Get the decimal approximations:
Jeff Bivin -- LZHS

11. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec.
If the ball lands in a 20 foot deep pit, when will the ball hit the bottom of the pit?
What height?
h(t) = -20
200 ft
20 feet below the ground
Jeff Bivin -- LZHS

12. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec.
If the ball lands in a 20 foot deep pit, when will the ball hit the bottom of the pit?
6.972 sec.
What height?
h(t) = -20
Get the decimal approximations:
Jeff Bivin -- LZHS

13. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec.
What will be the height of the ball in 3 seconds?
296 ft.
What time?
t = 3
Jeff Bivin -- LZHS

14. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec.
How far from the building will the ball land?
Wait !!!!
This formula compares time with height, not horizontal distance.
200 ft
Answer:
we don’t know! ?
Jeff Bivin -- LZHS

15. That was FUN!
Let's do one more!
Jeff Bivin -- LZHS

16. A diver dives off a 3 meter diving board into a pool with an initial upward velocity of 3.5 m/sec.
What is the maximum height of the diver?
When will the diver reach his/her maximum height?
When will the diver splash into the water?
What will be the height of the diver in 1 second?
Jeff Bivin -- LZHS

17. A diver dives off a 3 meter diving board into a pool with an initial upward velocity of 3.5 m/sec.
What is the maximum height of the diver?
When will the diver reach his/her maximum height?
When will the diver splash into the water?
What will be the height of the diver in 1 second?
3.625 meters
0.358 sec.
1.217 sec.
1.6 meters
Jeff Bivin -- LZHS

18. That was FUN!
Let's do one more!
Jeff Bivin -- LZHS

19. VERTEX Income = Price ● Quantity f(x) = ( 15 + 2x ) ( 200 – 10x )
A taxi service operates between two airports transporting 200 passengers a day. The charge is $ The owner estimates that 10 passengers will be lost for each $2 increase in the fare. What charge would be most profitable for the service? What is the maximum income?
VERTEX
Income = Price ● Quantity
Define the variable
x = number of $ price increases
f(x) = ( x ) ( 200 – 10x )
f(x) = 3000 – 150x + 400x – 20x2
f(x) = income
f(x) = – 20x x
f(6.25) = – 20(6.25) (6.25)
f(6.25) =
Vertex is:
So, price = (15 + 2x) = (15 + 2(6.25)) = = $27.50
Maximum income = f(x) = $
Jeff Bivin -- LZHS

20. That was FUN!
Jeff Bivin -- LZHS