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A wire is stretched from the ground to the top of an antenna tower. The wire is $20 \mathrm{ft}$ long. The height of the tower is 4 ft greater than the distance $d$ from the tower's base to the bottom of the wire. Find the distance $d$ and the height of the tower.

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Distance $d: 12 \mathrm{ft} ;$ tower height: $16 \mathrm{ft}$

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Missouri State University

Campbell University

Oregon State University

University of Michigan - Ann Arbor

So in this problem we have a right triangle untended tower, forming a right angle with the ground. What you are given is that the wire stretches from the top to the ground. At 20 feet, that is the high pot news. The tower itself is the leg that goes from the top of the high pot news down to the ground, and the ground itself is the second leg. Now the information you're given is that the height of the tower is for greater than the distance of the cowers base to the wire. So what we have to establish is what to call the distance from the bottom of the tower to where the wire meets the ground right there and in the directions. It says, Let's call it D. So we'll go ahead and give it the variable D. Now we can create the height of the tower, knowing that the ground is D. So the height of the Chandler is right there in the problem states that the tower is four feet more than the base, so there's four feet more than the base. This is by no meal, good practice wrapped by no meals in parentheses because if ever we have to square that, that becomes a little bit more of a complex issue. Where the right angle is is where the tower meets the ground right there. So the high pot news is obviously 20 in this case, and the two legs are the red and the green, both involving deep. We will solve for D by using path, agree and serum, which states the sum of the square of the two legs of a right triangle will always equal the square of the high pot. News he will sign is a little bit if he there. So let me create that equal little bit better a squared plus B squared to see square. We have a new addition of two legs. Edition is communicative, which means the order of the legs do not matter. You could use D for a or you may use D plus four for a You may use D for B, or you may use D plus four for be. What you don't have. A decision in is the fact that C must be 20. It must be the high pot news, so let's go ahead and substitute reading left to right. I see The D plus four is my first leg, so I'm just going to choose to make my d plus four be substituted where my A is right here. Rapid in parentheses. This green leg D plus four is not as simple as D squared, plus six teeth. It is not D squared plus 16 when it squared. You must distribute or foil this, which means you will be getting four terms. When this is multiplied together, you will be getting the D times, the d, the four times the d another four times D and a four times of four. So we have to be very careful about squaring a binomial. I'm gonna substitute my d the red leg for my B and that will equal my substituted. See, Will be 20 squared. All right, so now we're ready to excel for D. But let's clean it up before we continue. So I'm going to where my by no meal of depots for de Times D is the 1st 2 terms is D squared then My next distribution is deed Times four for D, and my next distribution after that is four times D. It's another 40. So put those two forties together and you get eight D and four times four is 16. There is my foiled D plus four leg. I'm going to keep my d squared B squared. I'm not gonna do anything with him right now, and I'm going to go ahead and square my 20 which is 400. Okay, we have a quadratic meaning we have D squares. We have a variable to the square power, which is a quadratic it graphs is a parabola. And in order to solve for for a quadratic we factor factoring means one side becomes a zero and the other side factors. So in this case, the right side is going to become zero. I'm going to take the 400 move it to the other side at the same time, I'm going to combine my life terms. I have a d squared and I have another d squared right here. So on the left side, I'm going to combine my D squares and that'll give me a to D squared, and I'm going to go ahead and move the 400 by subtraction, turning it into a zero on the right gonna move him to the other side. I'm gonna line that minus 400 up right under the 16. Cause those air, my Constance, those air my life terms. Those were the ones that can be combined. So I'm going to do all this in one step. You may take many steps as needed to do what I'm about to do. I am taking some shortcuts. I'm going to combine my both of my D squares and I get one d squared plus one d squared. And that is Judy Squirt. Um, my next in line of descending order is my D. So I'm gonna rewrite my 80. He doesn't combine with anybody, and then I am going to do my 16 minus 400 which is, um, minus 384. Sorry, it took me a minute to do the subtraction there. I put. Plus, I'm just gonna throw in a negative there because I kind of jumped the gun and put Plus, that's okay. Plus negative is the same as subtraction equals zero. Now we're ready to factor this probably no meal. I have my left side and descending order. Meaning my highest exponents first, followed by my next exponents power followed by the next, and it all equals zero. When you are factoring appalling, no meal you more than likely are going to get to set of polynomial factors, However, always, always make your life simple. See if there is a G c F. Since I have unequal sign, I can take a G C f and divide every single term by it. And that g cf is too. Why not make your life easy? The sooner you can get a positive one in front of the squared term, the easier your life is gonna be. Now that's not valid. What I just did, what you have to do is say, Oh, I also have to divide the right side every turn by two. Yes, it still is a zero, but you to make math ballot that zero must be shown as being divided by two as well. All right, you divided by two is positive, Juan. So that's what's in front of my D squared eight. Divided by two is four. That's what sits in front of my linear deed and negative 384 divided by two is 192. I'll just go ahead and make it minus 1 92 You can put plus negative 192. It's up to you, whichever is your easiest way to make sense out of this. All right, it's a quadratic again. We have a square. Do you turn now? We're ready to factor. We've gotten rid of our GC up of two. So we're getting ready to create Thies to Polly. No meal. In this case, binomial factors equal to zero. So you you look at the 192. It's negative. What combination Of two factors. One will have to be positive. One will be negative. Multiply to negative 192 that also add to four. Now, there are a lot of new mock devices out there that help you get there very, very quickly. But this is a word problem. You've gone through all those skill sets in order to factor relatively quickly. So I'm going to tell you right now what those two factors will be. We do start by taking the first turn d squared, breaking him up into the first term of both by no meals. Now I can't automatically put plus minus because I don't know what's gonna work, but what does Work is positive 16 times. Negative 12. That tells me positive. 16 times Negative 12 is negative 192 which it is, and positive 16 plus negative 12 is positive floor, which it ISS. Now we apply the zero product properties of which states, if you have any amount of factors equal to zero, all it takes is one of those factors being a zero to make this work. So we're gonna play out different scenarios. Factor one ISTEA plus 16. My first scenario is, if d plus 16 become zero, it doesn't matter what the other factors are. I know the left side will become a zero and make this truth. So I'm gonna play that scenario first and say, if d plus 16 0 I have a true statement. So let's go ahead now and solve that. We subtract 16 from both sides and we find that deace negative 16. Now we recall that the word problem is asking for a length. Our height of the tower does negative. 16 Makes sense. No, the tower is not underground. So even though that's a legitimate answer, it's not a valid answer for this particular word problem. So we show that we got it. But we throw it out saying the legitimate legitimacy of this being an actual answer in this word problem does not exist. Now we go to the second factor of zero product property says, if any factor equal zero, this whole left side become zero. So are other scenario is what if d minus 12 equals Europe. That will make the whole left side true. So let's figure this out. We'll add 12 to both sides and we find that D is 12. That makes sense. We know that D is 12 feet, so let's go back up to the problem itself and we realize we saw D, which is the red leg to be 12 feet now what's the length of the other? Now? We have permission to substitute well, where the D is right there, because the red leg told me D is now 12 and we find that the height of the tower is 12 plus four 16 feet. If you are asked to write your answer in a complete sentence, you would right. Well, I need to relook at the question. Actually, we're finding just the distance D from the height of the tower. And what I did is I went one step further and found the height of the tower re reading the question. We realized they didn't even ask for the height of the tower. They just asked for D. So your complete sentence would be the distance from the height of the tower to where the buyer meets. The ground is 12 feet.