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Problem 51 Hard Difficulty

A telephone line hangs between two poles $ 14 m $ apart in the shape of the catenary $ y = 20 \cosh (x/20) - 15, $ where $ x $ and $ y $ are measured in meters.
(a) Find the slope of this curve where it meets the right pole.
(b) Find the angle $ \theta $ between the line and the pole.

Answer

A. $\approx 0.3572$$\\$
B. $\approx 70.34^{\circ}$

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Vpxcbfxvn G.

October 20, 2020

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Video Transcript

So in this problem we have that a total line hangs between two poles that are 14 meters apart. So in party way have that the equation of the total mind pose. Why coat? Why equals 20 clothes on h of X over 20 minus 15. So, party, we need to find the slope of the telephone line along with the right hole. So how do we do this? Well, in order to find spoke, we need to find the derivative. And in order to do this, we need to basically put the equation of why, in terms of X and Y only and Garrett of the age. So if you remember from your textbook, we know that the co sign of age equals each of the X plus e to the negative X divided by two. However, in our case, we need to know that X is actually equal to X over 20. As you can see in this box right here in the Red King. So what does this mean? We can basically push altogether The equation. Why equals 20 times E to the X over 20 plus e to the negative parentheses. X over 20. And while we're starting out. It's important to put the parentheses, since this is a negative of the whole exciting, not just Dex. So this is all divided by two minus 15. So we have our final question. However, we can go one step further and typified by saying that 20 divided by two equals 10. So now that we have our final equation, we can put this into derivative. Now, if you know how to take a derivative, it's basically taking the 10 as a constant, multiplying it by the to the ex over 20 times, one over 20. Since the derivative of the exponents is won over 20 plus e to the negative X over 20 times negative one over 20 as follows by the previous each that's over 20. And all of this, we can say, is equal to the Y Prime Durga. No, what? What we can do further with this is that we know that the X value has to be seven, as you can see in this drawing, because the X coordinate on the right poll is seven. So we could basically plug in seven for ex into the derivative quite derivative equation here to get 0.3572 as be slow for a now let's move on to heart. Be so in part B were given that we need to know what the fada Askew contain. This drawing of the black angled right here it's so how can we do this? Well, in this problem, we can basically see that the bull angle right here is equal to 90 degrees. Namely, it's a right triangle so we can use that knowledge and see that there's a complimentary and gold to fado or the black black angle. And this is the red angle. Let's name it data to this problem. All right, So, as you know, from all right angles, we know that 90 minus satar to the right angle equals data because of the complimentary property of right angles. So what we can do is take the slope from part A, which was equal to sign a chit of seven over 20 due to the derivative of co signed H seven over 20 being sign age seven over 20. So we can take that hard a and say that the slope or qianjin of data too, or the red and over here is equal to that slope found in part. So when you rearrange everything, you essentially get that data to equals. The inverse tangent of sign. Age of seven over 20. Teo, isolate date or two. So now that we have our data to angle, we can basically say that we have all the person needed to find data according to this equation right here. So what are we gonna do? We're basically going to you put that data equals 90 degrees minus the inverse tangent of signed H of seven over 20. And this is all one section that indicated as stated to right here. So this is all data, too. Now it's your American part A you know, Dutch. The party a signed H of seven over 20 or the slope that we found was 200.3572 So what you can do is put in your graphing calculator in degrees mode, since we're using the 90 degrees instead of fly over to which would be used in radiance mode and new 92 90 degrees, minus the inverse tangent of side edge of seven over 20 or the inverse tangent of 200.357 too and this would equal 70.34 degrees or data. So that's the answer to party, and that's how you do the question.