Enlarging the Penguin PondDate: 06/03/99 at 14:14:11 From: Crystal Subject: Quadratic equation Hi! I'm not at word problems; they seem easy but they are hard for me to solve. Here is the problem: The rectangular penguin pond at the Bay Park zoo is 12 meters long by 8 meters wide. The zoo wants to double the area of the pond by increasing the length and width by the same amount. By how much should the length and width be increased? Date: 06/07/99 at 12:39:20 From: Doctor Arthur Subject: Re: Quadratic equation Crystal, Don't feel bad about word problems; most people don't do well with them. I suggest that the best way to do this problem would be with a chart so that you can see why they give you certain information. Here: --------------------------- Since the zoo wants to increase |Length|Width| Area | both the length and the width by --------------------------- same amount, we'll use the variable Now | 12 | 8 | 96 | x to represent the amount being Later| 12+x | 8+x |(12+x)*(8+x)| added to both. --------------------------- (Note: Since x is a length, x>0.) Since we know that the area later will be twice the current area, we can write the following equation: (12+x)*(8+x) = 2(96) By multiplying out and getting everything on one side, we get x^2 + 20x - 96 = 0 which factors into (X+24)(x-4) = 0 Using the zero product property, x = -24 and x = 4 Since x can't be negative, therefore x = 4. So if the zoo adds 4 meters to the length and the width, the area of the new pond would be double the area of the original. I hope this will help you out. Come back again if you need more. - Doctor Arthur, The Math Forum http://mathforum.org/dr.math/ Date: 06/03/99 at 17:00:56 From: Doctor Rob Subject: Re: Quadratic equation Thanks for writing to Ask Dr. Math! Here are some tips on solving "Word Problems." -------------------------------------------------------------- Solving an Applied Problem First convert the problem into mathematics. This step is (usually) the most challenging part of an applied problem. If possible, start by drawing a picture. Label it with all the quantities mentioned in the problem. If a quantity in the problem is not a fixed number, name it by a variable. Identify the goal of the problem. Then complete the conversion of the problem into math, i.e., find equations that describe relation among the variables, and describe the goal of the problem mathematically. Solve the math problem you have generated, using whatever skills and techniques you need. As a final step, you should convert the answer of your math problem back into words, so that you have now solved the original applied problem. --------------------------------------------------------------- Let's apply these steps to the zoo problem. We start by drawing a picture: G F o-----------------------------o | | | C | D o-----------------------o | | | | | | | | | | | | | o-----------------------o-----o A B E Here rectangle ABCD is the original pond. The problem tell us that AB = CD = 12 meters, and BC = AD = 8 meters, so we label those parts with those dimensions: G F o-----------------------------o | | | C | D o-----------------------o | | 12 | | |8 8| | | | | | 12 | | o-----------------------o-----o A B E Then we are told that the zoo wants to double the area. We know the formula for the area of a rectangle is A = L*W, where L is the length and W is the width, so we can find the area of the original pond: A = 8*12 = 96 square meters. The new pond will be longer by an amount we don't know but want to find. Let's call that x meters, following the idea of naming quantities we don't know with variables. That means that the new pond length AE = FG = 12+x meters. We are also told that the width of the pond is increased by the same amount, that is, x meters, so the new pond width is 8+x meters. We label BE and DG with x, and EF with 8+x and FG with 12+x: G 12+x F o-----------------------------o x| | | C | D o-----------------------o | | 12 | |8+x |8 8| | | | | | 12 | x | o-----------------------o-----o A B E Now there is one last part of the problem statement: the part which will give us an equation to solve. That is, that the area of the new pond (which we know is its length times its width) is double that of the old pond (which we figured was 96). Here is the equation: (12+x)*(8+x) = 2*96. Now we have finished translating the problem into mathematics and we use the appropriate math technique to solve this equation. In this case, when you multiply everything out (using FOIL, for example) and collect like terms, you will have a quadratic equation. Choose your favorite way to solve quadratic equations and find the values of x which make this equation true. There are two, because quadratic equations always have two solutions. Now we have to translate the answers back into a solution of the word problem. One of the values of x will be negative, and that will make no sense in this problem, so that value of x should be discarded. The remaining solution will be given in the form, "The zoo should make the pond __ meters longer and wider." Fill in the blank with that value of x. Understand? The same techniques can be used on essentially all "Word Problems." - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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