penerapan grafik parabola
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Quadratic Applications-------------------------------
Vertical Motion&
Profit / Income
By: Jeffrey Bivin
Lake Zurich High School
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
80−
−=t
3280
−−=t ( ) 3002
5 =h
Vertex is: ( )300,25
300 ft.
25=t ( ))(, 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
80−
−=t
3280
−−=t ( ) 3002
5 =h
Vertex is: ( )300,25
2.5 sec
25=t ( ))(, 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: 830.1−≈t
6.830 sec.
830.6≈tJeff 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: 732.0≈t
0.732 sec &
4.268 sec.
268.4≈t
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: 453.1−≈t
6.453 sec.
453.6≈t
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 ++−= ttth
What 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 ++−= ttth
What height? h(t) = -20
200801620 2 ++−=− tt
( ) ( ))16(2
2201648080 2
−−−±−=t
322048080
−±−=t
Get the decimal approximations: 972.1−≈t
6.972 sec.
972.6≈t
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−
−=x25.6=x
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