Word Problems Involving Factoring [PDF]

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Word Proble ms In vol vin g



Fa c t o r i n g



Word Problems Involving Quadratics These word problems involve situations I've discussed in other word problems: The area of a rectangle, motion (time, speed, and distance), and work. However, these problems lead to quadratic equations. You can solve them by factoring or by using the Quadratic Formula. Example. One number is the square of another. Their sum is 132. Find the numbers. Let A and B be the numbers. The first sentence says one is the square of the other, so I can write The sum is 132, so Plug



into



and solve for B:



The possible solutions are and If , then . If , then . So two pairs work: -12 and 144, and 11 and 121.



.



Example. The difference of two numbers is 2 and their product is 224. Find the numbers. Let x and y be the numbers. Their difference is 2, so I can write Their product is 224, so From



, I get



. Plug this into



and solve for y:



If , then . If , then . So two pairs work: -14 and -16, and 14 and 16. Example. The area of a rectangle is 560 square inches. The length is 3 more than twice the width. Find the length and the width. Let L be the length and let W be the width. The length is 3 more than twice the width, so The area is 560, so Plug in



and solve for W:



Use the Quadratic Formula:



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Word Proble ms In vol vin g



Since the width can't be negative, I get



. The length is



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.



Example. Calvin and Bonzo can eat 1260 hamburgers in 12 hours. Eating by himself, it would take Calvin 7 hours longer to eat 1260 hamburgers than it would take Bonzo to eat 1260 hamburgers. How long would it take Bonzo to eat 1260 hamburgers by himself? Let c be Calvin's rate, in hamburgers per hour. Let b be Bonzo's rate, in hamburgers per hour.



The last equation gives



The second equation gives



Plug



Plug



The solution and Calvin takes



into



:



into the first equation



and solve for t:



doesn't make sense, since time can't be negative. The solution is 21. Bonzo takes 21 hours, hours.



Example. If Calvin and Bonzo eat together, they can eat 480 hot dogs in 6 hours.



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Word Proble ms In vol vin g



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Eating alone, Bonzo takes 16 hours longer than Calvin would to eat 480 hot dogs. How long does it take Calvin to eat 480 hot dogs? Let x be Calvin's rate (in hot dogs per hour), let y be Bonzo's rate, and let t be the time it takes Calvin to eat 480 hot dogs.



The second equation says



, so



The third equation says Plug these into the first equation



The solutions are



and



. , so



. and solve for t:



. Since t can't be negative, the answer is



hours.



Example. Calvin rides his power boat up and down a drainage ditch. The water in the drainage ditch flows at 6 miles per hour. Calvin takes 5 hours longer to travel 360 miles against the current than he does to travel 360 miles with the current. What is the speed of Calvin's boat in still water? Let x be the speed of Calvin's boat in miles per hour in still water, and let t be the time in hours it takes him to travel 360 miles with the current. Thus, it takes him hours to travel 360 miles against the current.



The equations are Solve the second equation for t: Plug this into the first equation and solve for x:



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Word Proble ms In vol vin g



The solutions are



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. Since the speed can't be negative, the answer is 30 miles per hour.



Example. The hypotenuse of a right triangle is 4 times the smallest side. The third side is hypotenuse and the smallest side. Let s be the smallest side and let h be the hypotenuse. By Pythagoras,



. Find the



The hypotenuse is 4 times the smallest side, so Plug



into



and solve for s:



Since doesn't make sense, the solution is . Then A box with no top is to be made from a piece of metal (#3510)



.



Solutions for Solving Word Problems using Factoring 1. The product of two consecutive integers is 272. Find the value of each integer. The first thing you need to do is to define the integers. Let n = the first integer Let n+1 = the 2nd integer The product means to multiply so we need to multiply the two integers together. (First integer)*(Second integer) = 272 (n)*(n + 1) = 272 Now multiply everything out and set it equal to zero. n2 + n = 272 n2 + n -272 = 272 – 272 n2 + n -272 = 0 Now you need to factor and solve. (n + 17)(n – 16) = 0 n + 17 = 0 or n – 16 = 0 n = -17 or n = 16 Now you need to go back and answer the questions using each answer. If n = -17 then the 2nd integer is n + 1 or -17 + 1 = -16 so the two integers are -17 and -16 If n = 16 then the 2nd integer is n + 1 or 16 + 1 = 17 so the two integers are 16 and 17.



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Word Proble ms In vol vin g



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Remember that negative answers are not always bad. 2. The product of two consecutive even integers is 528. Find the value of each integer. We are going to follow the same steps as in #1. Let n = the first integer Let n+2 = the 2nd integer (First integer)*(Second integer) = 528 (n)*(n + 2) = 528 n2 + 2n = 528 n2 + 2n -528 = 528 – 528 n2 + 2n -528 = 0 (n + 24)(n – 22) = 0 n + 24 = 0 or n – 22 = 0 n = -24 or n = 22 If n = -24 then the 2nd integer is n + 2 or -24 + 2 = -22 so the two integers are -24 and -22 If n = 22 then the 2nd integer is n + 2 or 22 + 2 = 24 so the two integers are 22 and 24. 3. A rectangular swimming pool is twice as long as it is wide. A small concrete walkway surrounds the pool. The walkway is a constant 2 feet wide and has an area of 196 square feet. Find the dimensions of the pool. The first thing you need to do is to define your variables. In this example, the length is described in terms of the width so define the width. Let w = the width of the pool The problem says that the length is twice as long as the width and remember that twice means to multiply by 2. Let 2w = the length of the pool This means that the area of the pool is A = (w)(2w) The pool has a concrete walkway of 2 feet surrounding it so this means that there is 2 feet on EACH SIDE of the width and the length so we need to increase BOTH the length and width by 4 in order to get the length and width of the pool and the walkway. This means that the width is w + 4 and the length is 2w + 4. This means that the walkway and pool area is: A = (w + 4)(2w + 4) We are also given that the area of the walkway and the pool is 196 square feet.



In this case we have two areas. We have the area of the pooland the area of the walkway and pool (See picture). In order to find the dimensions we need to use the two different areas. Area Total = Area Walkway and Pool – Area Pool 196 = (w + 4)(2w + 4) - (w)(2w) 196 = (2w2 + 4w + 8w + 16) – 2w2 196 = 2w2 + 4w + 8w + 16 – 2w2



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Word Proble ms In vol vin g



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196 = 12w+ 16 196 - 16 = 12w+ 16- 16 180 = 12w 12 12 15 = w So now we know that the width of the pool is 15 ft. But we aren’t done yet. The question asked for the dimensions so we still need to find the length. The length is twice the width so 2 * 15 = 30ft. The dimensions of the pool are 15 ft by 30 ft. In a right triangle, the longer leg is 10 more than the longer leg. If the hypotenuse is 50, what are the measures of the two legs. a



+b



x



+(x+10)



x



+(x+10)(x+10)=2500



x



+x



2x Since the sides of a triangle are always positive, the short leg must be 30 and not - 40. Therefore, the other leg is x + 10 = 30 + 10 = 40



=c =50



+20x+100=2500



+ 20x - 2400=0



x +10x -1200 = 0 (x + 40)(x + -30)=0 x + 40 = 0 or x - 30 =0 x = -40 or x = 30