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MG
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Problem 33 Medium Difficulty

Archaeologists have discovered that the 18 th-century garden of the Charles Carroll House in Annapolis, Maryland, was a right triangle. One leg of the triangle was formed by a 400 -ft long sea wall. The hypotenuse of the triangle was $200 \mathrm{ft}$ longer than the other leg. What were the dimensions of the garden?
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Answer

300 ft by 400 ft by 500 ft

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Video Transcript

this particular problem. We are looking at a right triangle, which means Pythagorean serum is gonna come into play. Now what is given on this right triangle is ah, 400 foot leg right here. In this problem, they make reference to the fact that the high pot news is 200 more than the other leg. We can't identify the high pot news until we define the other leg. So what we're going to do is create a name for the other leg. More than likely, you will use X variable. You can choose any variable to represent that leg, but more often than not, we use an X. So that's what I'll use. And once I identify the leg, then I can make use of the information that the high pot news is 200 more than the other lay, which is X. Here is our right angle. So X plus 200 is the high pot news breath agree of serum states that the sum of the square of two legs of a right triangle We'll always equal the square of the high pop news. You have no choice. But that see will be substituted with experts 200 the A and B can be inter changed because edition is communicative between the X and the 400. It's your choice because I read left to right. I'm gonna substitute the X in for my because that's the first leg that I see left to right. But again, it makes no difference. And I'm going to substitute the 400 where the bs If you do 400 scared plus X squared, you're totally 100%. Okay. What we have to keep in mind is how careful we need to be with this high pot Knicks. What we see here is a buying no meal right here. I'm gonna wrap this in parentheses by no meal squared is not this particular binomial will not be X squared plus 40 demos. It no, this must be foiled. X plus 20 in parentheses, times X plus 20 in parentheses and foiled. So I'm gonna put that in parentheses to ensure that he gets foiled and foiling means this is X plus 20 time at 200 plus x plus two times x plus 200. I'm going to rewrite the right son. Gonna scoot over a little bit to make room on my page plus 400 squared is oops, my bad. There it is the race that 320,000 x plus 200 the quantity squared. My first term is X Times X, which is X Square. The second terms will be the X Times 200 which is 200 X. The third term is 200 times acts, which is 200 X. I combine those two terms together and I will get a 400 X So 400 axes, a result of 200 X plus 200 X. Then I multiply my last terms 200 times 200 and get 40,000. So what? We're staring at us, Upali. No meal, multiple terms on both sides of the equal Sign a quadratic because access square. So the goal is to make one side completely a zero and bring all like terms over to the other side and combine like terms so that you have a polynomial equal to zero, and then we begin the factoring process. So it doesn't matter which side you choose to take to which side. Um, I am going to take my right side to join my left side. So I'm going to subtract out my ax squared and take him over here. Interesting. My X squared is going to kind of disappear. I'm going to remove my 400 X from this side. I'm gonna kind of sneak him right here. There is no, like term over on the left to combine him with, so he's going to stand out on his own. And then I am going to subtract my 40,000 and he is going to combine with the 160,000 on the left side and always equal to each other. So the left is totally equal to the right. All of these on the right just became a zero, which is the goal in factoring. This happened to cancel out, which just made her life a little bit easier. The minus 400 acts has no one to combined with, So he sits on his own as a negative. 440,000 acts, 160,000 minus 40,000 is a plus 120,000. Now this just became linear. What you see is we no longer have the X squared so factoring has just gone away and we're left with the linear axle we solve by bringing everything except acts to the other side of the equation. So it looks like I'm gonna have to undo what I just did a little bit, but that's OK. We start by taking the constant to the right by turning him into a zero on the left side. But what we do to the last sign we will do to the right side. It looks a little weird because we have all these negative signs, but you realize it will work itself out in the end. So what we just achieved with the additive identity over here of zero, we have a negative 40,000 x plus zero in essence, over on the left. But the plus zero as an additive identity does no longer need to be seen. And we have a negative 120,000 over on the right side. Now we see it come into play that we can divide out the negative 40,000 because what we're trying to get here is a multiplication of identity of positive one negative, 40,000 divided by negative 40,000 is in fact, positive one and only when we have a positive one in front of the acts being multiplied by acts, do we consider acts to be solved so defining. Or you can choose to multiply by the reciprocal of negative 40,000 which is times negative one over 40,000. Multiplying by the reciprocal is exactly the same as division. This becomes a positive one x I can choose to delete or not have the positive would not be seen because it's an identity. And over on this side we have 30,000 as any answer. So the question was asking, What are the three dimensions we just found? The acts is equal to 300 feet. That makes this leg 300 feet, cause the leg was identified as acts. It makes the high part news. We're gonna substitute 300 for acts. So now we see that the high pot news is 300 plus 200 500 feet. If you are asked to answer in a complete sentence, you would answer this problem as the dimensions of this triangle or 300 feet by 400 feet by 500 feet

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