Quadratic Applications-------------------------------

Vertical Motion&

Profit / Income

By: Jeffrey Bivin

Lake Zurich High School

jeff.bivin@lz95.org

Last Updated: November 30, 2007

Vertical Motion

• Compares the height of an object with the time in flight.

002

21)( htvgtth

g = force of gravity: 32ft/sec or 9.8 m/sec

vo = initial velocity

ho = initial height

Jeff Bivin -- LZHS

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

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.

We need to use: 002

21)( htvgtth

20080)32()( 221 ttth

g = 32 ft/s2

vo = 80 ft/s

ho = 200 ft

2008016)( 2 ttth

• What is the maximum height of the ball?

Jeff Bivin -- LZHS

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?

2008016)( 2 ttthWhere is the maximum? Find the vertex……

abt 2

2008016 252

25

25 h

2008016 25

425

25 h)16(2

80t

3280

t 3002

5 h

Vertex is: 300,25

300 ft.

25t )(, tht

Jeff Bivin -- LZHS

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?

2008016)( 2 ttthWhere is the maximum? Find the vertex……

abt 2

2008016 252

25

25 h

2008016 252

25

25 h)16(2

80t

3280

t 3002

5 h

Vertex is: 300,25

2.5 sec

25t )(, tht

Jeff Bivin -- LZHS

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?

2008016)( 2 ttthWhat is the height at the ground? h(t) = 0

20080160 2 tt

)16(2

2001648080 2

t

321920080

t

Get the decimal approximations: 830.1t

6.830 sec.

830.6tJeff Bivin -- LZHS

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.

2008016)( 2 ttth• When will the ball be at a height of 250 feet?

250 feet What height? h(t) = 250

2008016250 2 tt

)16(2

501648080 2

t

32320080

t

Get the decimal approximations: 732.0t

0.732 sec &

4.268 sec.

268.4t

5080160 2 tt

Jeff Bivin -- LZHS

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?

2008016)( 2 ttthWhat height? h(t) = 400

2008016400 2 tt

)16(2

2001648080 2

t

32640080

t

Wait, what was the maximum height?

never

ft300

20080160 2 tt

Jeff Bivin -- LZHS

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?

2008016)( 2 ttthWhat height? h(t) = 50

200801650 2 tt

)16(2

1501648080 2

t

321600080

t

Get the decimal approximations: 453.1t

6.453 sec.

453.6t

15080160 2 tt

Jeff Bivin -- LZHS

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.

20 feet below the ground

• If the ball lands in a 20 foot deep pit, when will the ball hit the bottom of the pit?

2008016)( 2 ttthWhat height? h(t) = -20

200801620 2 tt

Jeff Bivin -- LZHS

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?

2008016)( 2 ttthWhat height? h(t) = -20

200801620 2 tt

)16(2

2201648080 2

t

322048080

t

Get the decimal approximations: 972.1t

6.972 sec.

972.6t

22080160 2 tt

Jeff Bivin -- LZHS

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?

2008016)( 2 ttthWhat time? t = 3

200)3(80)3(16)3( 2 h

296 ft.

200240144)3( h

296)3( h

Jeff Bivin -- LZHS

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?

2008016)( 2 ttthWait !!!!

Answer: we don’t know!

This formula compares time with height, not horizontal distance.

Jeff Bivin -- LZHS

Jeff Bivin -- LZHS

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

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

Jeff Bivin -- LZHS

A taxi service operates between two airports transporting 200 passengers a day. The charge is $15.00. 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?

Jeff Bivin -- LZHS

Income = Price ● Quantity

f(x) = ( 15 + 2x ) ( 200 – 10x )

Define the variable

x = number of $2 price increases f(x) = 3000 – 150x + 400x – 20x2

f(x) = – 20x2 + 250x + 3000

VERTEX

abx 2

)20(2250

x25.6x

f(6.25) = – 20(6.25)2 + 250(6.25) + 3000

f(6.25) = 3781.25 Vertex is:

25.3781,25.6

So, price = (15 + 2x) = (15 + 2(6.25)) = 15 + 12.5 = $27.50

f(x) = income

Maximum income = f(x) = $3781.25

Jeff Bivin -- LZHS