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Problem 31 Medium Difficulty

Elliott Street is $24 \mathrm{ft}$ wide when it ends at Main Street in Brattleboro, Vermont. A 40 -ft long diagonal crosswalk allows pedestrians to cross Main Street to or from either corner of Elliott Street (see the figure). Determine the width of Main Street.
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Answer

$32 \mathrm{ft}$

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Algebra

Elementary and Intermediate Algebra

Chapter 6

Polynomial Factorizations and Equations

Section 7

Applications of Polynomial Equations

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Video Transcript

So this problem is talking about a right triangle. Once again, you are given Ah, hi. Pot knows. Ah, 40 feet along with a leg of 24 feet. So that's your given information. And you are asked to find the width of this particular street. That is X. So we create a right triangle right here, and we're going to then use Pythagorean steer. Um, in order to complete the other side. The thuggery and serum states a squared plus B squared equals C squared. Now A and B are the two legs of the right triangle. Those were the two legs that are adjacent to hurt me that are adjacent to the right angle. The hypothesis 40. It is opposite the right angle. So we will plug in either acts or 24 for a and either X or 24 for be. It does not matter. Their two legs and addition is communicative. So the order you plug them and does not matter because I'm reading left to right. I see the X first, I'm going to choose to plug in my axe for the A again. You could put 24 there if you choose to and I'm gonna put 24 in for the B. This this substitution, and I'm gonna put in substitute my 40 form. I see. Okay, now I'm going to start solving for X. I do have some clean up to do, and namely, that means squaring my 24 squaring my four x. I'm going to do that right now. I still have ax squared. 24 square is 576 and 40 square is 1600. So there's just a step of cleaning up my squared numbers. We're in a factoring unit, so I'm gonna choose to do factoring, factoring means bring all terms to the left side. The right side equals zero. So that's what I'm going to do right now. I'm going to move the 1600 to the left side so that this equation now equals zero and is ready to be factored. So when I do that, I get X squared. 576 minus 1600 is 1024. 1000 24 and now it's equal to zero. So now it's ready to be factored. So factoring means, um, taking the left side and bringing it down into factors that we can then apply the zero product property too. This particular Polly No meal is a difference of two perfect squares. In order to apply the difference of two perfect squares, you must have a minus sign which we dio and you must have to perfect square terms on either side of the minus sign which we dio when you have a difference of two perfect squares. You have to by no meal factors where the first term is going to be the square root of X squared. So the first term is gonna be the X status. A square root of X squared. One of the terms gets a plus sign. The other term will get a minus sign again. Order doesn't matter. Multiplication is communicative and factors are multiplication. So if you put the minus in the first factor in the plus in the second factor, you are 100% okay with that, the next step is to take the 1024 give me it's perfect square root which is 32. And so the 32 goes as the second term in these by no meals. One following the plus sign one following the minus sign. Now we have two factors. Access to the first power in both of them, which is what we're trying to seek equal to zero. The zero product property states that if you have factors equal to zero, any factor when it becomes equal to zero will make this truth. So we're gonna play out two different scenarios we're going to say, Well, factor one, this is factor one. If it equals zero, then this whole statement is true regardless of what the second factor is equal to. So we're going to solve our first equation. We're gonna have to equations, subtract the 32 and we find out that one of our solutions is ax equals negative. 32. Okay, there's another alternative that says the second factor. If that's equal to zero, here's your second factor. If he's equal to zero, then so is the entire left side. So we're gonna play out the second scenario which says, Oh, factor to let's make him equal to zero and 32 to both sides, and we find that our second solution is 32. Now we have two solutions. They're both really solutions to this problem. But only one of them makes sense to the problem. Story problems. You always have to go back and read the question. Figure out what it is we're looking for. We're looking for the length off a street right here acts is the length the width of Main Street. Is there such a thing as a negative 32 with no. So we do throw out one of the answer's Yes, it's a really answer that exists in this probably no meal. But it's not applicability to the word problem. So this is the on Lee, one of the two answers that makes this make sense. So the answer to this word problem is 32 we always label feet. The should have been able to now that I look at that. So there you go beautifully done.

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