Applications of Quadratic Equations

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?

GEOMETRY

Let's draw a picture: L = w + 3

Call the width w w

Area = length x width so

10 = Lw = (w+3)w Solve this by multiplying out and getting everything on one side = 0 and factoring

0 = w2 + 3w - 10

-10 -10

0 = (w +5)(w-2)

w + 5 = 0 or w – 2 = 0

w = -5 or w = 2

Since width can't be negative throw out –5 and width is 2 m

L = 2 + 3 = 5 m

2 m

5 m

P = $500 and A = $572.45

COMPOUND INTEREST

Let's substitute the values we are given for P and A

Solve this equation for r

21 rPA Amount in account after two years

Principal Amount you deposit

Interest rate as a decimal

500500 2150045.572 r

211449.1 r

Square root both sides but don't need negative because interest rate won't be negative

r 107.1 %707.107.1 r

PYTHAGOREAN THEOREM

An L-shaped sidewalk from building A to building B at St Stephen’s 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? Draw a picture:

x

200-x

If first part of sidewalk is x and total is 200 then second part is 200 - x A

B

150

222 cba

Using the theorem: 222 150200 xx Multiply out

2250040040000 22 xxxcontinued on next slide

1

2250040040000 22 xxx get everything on one side = 0

0175004002 2 xx divide all terms by 2

087502002 xx use the quadratic formula to solve

12

875014200200 2 x

225100 x 6.64or 4.135 xx

200 - 134.5 = 64.6 so doesn't matter which you choose, the two lengths are 135.4 metres and 64.6 metres.

WORK-RATE PROBLEMAn 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

Time to complete

job

Rate for B

1 over time to complete alone

Time to complete

job1+ =

Call t time for machine A to

complete

1812

18

1

tt

(continued on next slide)

112

88

tt

Clear the equation of fractions by multiplying all terms on both sides by the common denominator and cancel all fractions.

12112

128128

ttt

tt

t

tt

tttt 128968 2 Get everything on one side = 0 and factor

09642 tt 0812 tt

So Machine A can complete the job alone in 12 minutes and Machine B would take 12 + 12 or 24 minutes.

8or 12 ttThrow out –8 because negatives don't make sense as a time to complete the job

After how many seconds will the height be 11 metres?

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.

,52116 2 tth

,5211611 2 ttGet everything on one side = 0 and factor or quadratic formula.

062116 2 tt-11 -11

162

61642121 2

t32

5721

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.

When will the tennis ball hit the ground?

,52116 2 tth

What will the height be when it is on the ground? h = 0

,521160 2 tt

162

51642121 2

t32

76121

So there are two answers: (use a calculator to find them)

t = - 0.21 or 1.52 seconds (throw out the negative one)

Average Speed

Let's make a table with the information

first part

second part

distance rate time

100

135

r t

r - 5 5 - t

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.

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?

first part

second part

distance rate time

100

135

r t

r - 5 5 - t

Distance = rate x timeUse this formula to get an equation for each part of trip

100 = r t 135 = (r - 5)(5 - t)Solve first equation for t and substitute in second equation

r r

r

100

rr

10055135

rr

10055135

FOIL the right hand side

rr

500251005135 Multiply all terms by r

to get rid of fractionsr r r r

r

05002605 2 rr Combine like terms and get everything on one side

Divide everything by 5 0100522 rr

Factor or quadratic formula

0250 rr

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.

2or 50 rr

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.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 CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au