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Applications of Quadratic Equations

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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 ww 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 = w 2 + 3w - 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

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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 Amount in account after two years Principal Amount you deposit Interest rate as a decimal 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 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 Using the theorem:Multiply out continued on next slide

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1 get everything on one side = 0 divide all terms by 2 use the quadratic formula to solve 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 Time to complete job Rate for B 1 over time to complete alone Time to complete job 1 += Call t time for machine A to complete (continued on next slide)

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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 So Machine A can complete the job alone in 12 minutes and Machine B would take 12 + 12 or 24 minutes. Throw out –8 because negatives don't make sense as a time to complete the job

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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. Get everything on one side = 0 and factor or quadratic formula. -11 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.

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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 Let's make a table with the information first part second part distanceratetime 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?

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first part second part distanceratetime 100 135 r t r - 5 5 - t Distance = rate x time Use this formula to get an equation for each part of trip 100 = r t135 = (r - 5)(5 - t) Solve first equation for t and substitute in second equation rr

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FOIL the right hand side Multiply all terms by r to get rid of fractions rrrr 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.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.www.mathxtc.com Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au

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