Length of a Walkway A nature conservancy grou...

Question

Length of a Walkway A nature conservancy group decides to construct a raised wooden walkway through a wetland area. To enclose the most interesting part of the wetlands, the walkway will have the shape of a right triangle with one leg 700 yd longer than the other and the hypotenuse 100 yd longer than the longer leg. Find the total length of the walkway.

Accepted Answer

Hello, today we're going to be solving the following problem. So we are told that a playground has a shape of a right triangle and the length of the shorter leg is seven m shorter than the longer one. Also, the high pont neues is nine m longer than the shorter leg. We want to calculate the length of the shorter leg. So let's go ahead and highlight a couple of things. We are told that the shape of the right triangle of the playground is a right triangle. So let's go ahead and draw a right triangle. Next, we are told that the length of the shorter leg is seven m shorter than the longer one. So we are not told what the length of the long leg is going to be. So we can represent that with the variable X. But we are told that the length of the shorter leg is 7m shorter than the longer one. And we can represent that by X -7. Next, we are told that the length of the hypotenuse is nine m longer than the shorter leg. And we know that the length of the shorter leg is being represented by X minus seven. So we're going to be adding nine to this quantity for the length of the hypotenuse. And that is going to give us X minus seven plus nine, which is going to give us the length X plus two. So now that we have the length for all three sides of the right triangle, we can calculate the length of the short leg by using the Pythagorean theorem. And the Pythagorean theorem states that the hypotenuse squared is equal to the sum of the side squared. And so we're going to let C be the hypotenuse, we're going to let a be the length of the long leg and we're going to let B be the length of the short leg. So plugging in the values is going to give us X plus two squared is equal to X squared plus X minus seven squared. Now, what we need to do is simplify and sulfur X and we can go ahead and start by foiling the two quantities. So let's go ahead and foil X plus two squared, X plus two squared can be rewritten as X plus two times X plus two. And in order to foil this, we're going to multiply X with X, which is going to give us X squared X with two, which is going to give us two X two with X, which is going to give us two X and two with two, which is going to give us four and combining like terms together is going to give us X squared plus four, X plus four. Now let's go ahead and foil X minus seven, X -7 Squared can be rewritten as X -7 times X -7. We're gonna multiply X with X which is going to give us X squared X with negative seven which is going to give us negative seven X multiply negative seven with X which is going to give us negative seven X and negative seven with negative seven, which is going to give us positive 49. And combining like terms is going to give us X squared minus 14 X plus 49. So let's go ahead and substitute these values in for the two squared quantities that's going to give us X squared plus four, X plus four is equal to X squared plus X squared minus 14 X plus 49. Now, on the right hand side, we can add the two X squirts together to give us two X squared. And since we want to solve for X, we want to set this equation equal to zero in order to do that, let's subtract all the terms from the left hand side to the right hand side and that is going to give us zero is equal to X squared minus 18 X plus 45. Now, on the right hand side, we have a factory trin Amiel as this trin Amiel is going to factor the quantities X -15 times X -3. And using the zero Theorem, we can set both of these factors equal to zero. So what we get is X -15 is equal to zero and we get X -3 is equal to zero. On the left hand side, if we add 15 to both sides of this equation, we get X is equal to 15. And on the right hand side, if we add three to both sides, we get X is equal to three. So we have the value of X that we can plug into the side length that we want to find. Now recall that we're representing the length of the short leg as X -7. So let's go ahead and plug in the values of X and test to see which one works with this value. So let's go ahead and test out X is equal to three. If X is equal to three, we get three minus seven, Which is going to give us -4. But X cannot be a negative value because a side length cannot be negative. So X -3 or X is equal three is not going to be a solution. Now, let's go ahead and test that when X is equal to 15, when X is equal to 15, we get 15, Which is going to equal to positive eight. And that means that the side length of the shorter side is going to be eight, which works. So that means X is equal to 15 is going to be our solution. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Key Concepts

Right Triangle Properties

Pythagorean Theorem

Algebraic Expressions and Equations

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