Applications of Quadratic Equations

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Applications of Quadratic Equations
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GEOMETRY
The top of a coffee table is 3 metres longer
than it is wide and has an area of 10 square
metres. What are the dimensions of the top of
the coffee table?
5 m
L w 3
Let's draw a picture
Call the width w
w
2 m
Area length x width so 10 Lw (w3)w
Solve this by multiplying out and getting
everything on one side 0 and factoring
-10
-10
0 w2 3w - 10
w -5 or w 2
L 2 3 5 m
0 (w 5)(w-2)
Since width can't be negative throw out 5 and
width is 2 m
w 5 0 or w 2 0
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COMPOUND INTEREST
Amount in account after two years
Interest rate as a decimal
Principal Amount you deposit
P 500 and A 572.45
Let's substitute the values we are given for P
and A
Solve this equation for r
500
500
Square root both sides but don't need negative
because interest rate won't be negative
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PYTHAGOREAN THEOREM
An L-shaped sidewalk from building A to building
B at St Stephens School is 200 metres long. By
cutting diagonally across the grass, students
shorten the walking distance to 150 metres. What
are the lengths of the two legs of the sidewalk?
200-x
Draw a picture
B
x
If first part of sidewalk is x and total is 200
then second part is 200 - x
150
A
Using the theorem
Multiply out
continued on next slide
5
get everything on one side 0
divide all terms by 2
use the quadratic formula to solve
1
200 - 134.5 64.6 so doesn't matter which you
choose, the two lengths are 135.4 metres and 64.6
metres.
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WORK-RATE PROBLEM
An office contains two copy machines. Machine B
is known to take 12 minutes longer than Machine A
to copy the company's monthly report. Using both
machines together, it takes 8 minutes to
reproduce the report. How long would it take
each machine alone to reproduce the report?
Work done by Machine A
Work done by Machine B
1 complete job


Rate for A 1 over time to complete alone
Rate for B 1 over time to complete alone
Time to complete job
Time to complete job


1
Call t time for machine A to complete
(continued on next slide)
7
Clear the equation of fractions by multiplying
all terms on both sides by the common denominator
and cancel all fractions.
Get everything on one side 0 and factor
Throw out 8 because negatives don't make sense
as a time to complete the job
So Machine A can complete the job alone in 12
minutes and Machine B would take 12 12 or 24
minutes.
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Height of a tennis ball
A tennis ball is tossed vertically upward from a
height of 5 metres according to the height
equation where
h is the height of the tennis ball in metres and
t is the time in seconds.
After how many seconds will the height be 11
metres?
So there are two answers(use a calculator to
find them making sure to put brackets around the
numerator)t .42 seconds or .89 seconds.
Get everything on one side 0 and factor or
quadratic formula.
-11
-11
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When will the tennis ball hit the ground?
What will the height be when it is on the ground?
h 0
So there are two answers (use a calculator to
find them) t - 0.21 or 1.52 seconds (throw out
the negative one)
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Average Speed
A truck traveled the first 100 kilometres of a
trip at one speed and the last 135 kilometres at
an average speed of 5 kilometres per hour
less. If the entire trip took 5 hours, what was
the average speed for the first part of the trip?
If you used t hours for the first part of the
trip, then the total 5 minus the t would be the
time left for the second part.
Let's make a table with the information
r
100
t
135
r - 5
5 - t
11
Use this formula to get an equation for each part
of trip
Distance rate x time
distance
rate
time
r
first part
100
t
second part
135
r - 5
5 - t
Solve first equation for t and substitute in
second equation
100 r t
135 (r - 5)(5 - t)
r
r
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FOIL the right hand side
r
Multiply all terms by r to get rid of fractions
r
r
r
r
Combine like terms and get everything on one side
Divide everything by 5
Factor or quadratic formula
So r 50 km/h since r 2 wouldn't work for
second part where rate is r 5 and that would be
3 if r was 2.
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Acknowledgement I wish to thank Shawna Haider
from Salt Lake Community College, Utah USA for
her hard work in creating this PowerPoint. www.sl
cc.edu Shawna has kindly given permission for
this resource to be downloaded from
www.mathxtc.com and for it to be modified to suit
the Western Australian Mathematics Curriculum.
Stephen Corcoran Head of Mathematics St
Stephens School Carramar www.ststephens.wa.edu.
au