Problem 50 Hard Difficulty

A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet (see the photo on page 839 ). The diameter at the base is $280 \mathrm{~m}$ and the minimum diameter, $500 \mathrm{~m}$ above the base, is $200 \mathrm{~m} .$ Find an equation for the tower.

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03:51

WZ

Wen Z.

02:46

Carson M.

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Video Transcript

in this problem. A cooling towel for a nuclear reactor is to be constructed in the shape of a hyperbole Lloyd of one sheet. Okay. The diameter at the base is to 80 m and the minimum diameter 500 m above the base is 200 meters. Find an equation for the travel. All right. So here it is. Given diameter. Mhm. At base it's what is the quest to 2 80 makers here? No minimum diameter. Yeah. Yeah. Yeah. 500 m above D. Best You think it was 2 200 m? No, assuming I'm sure. Mhm. We assume that yes. Yeah. The center to be okay. 000 that is for it. Okay. No. Yeah, equation all mm. Hyperbole Lloyd would be yeah, excess square by a square plus by square by B squared plus sees I'm sorry. This is very squared by C. Squared. It goes to one. Now from the given information. Mhm From D Yeah. Given information. We can say the horizontal cross section will be D hi design. Do prosection. Mhm busy certain. Yes. Yeah that is you can say it was to be here. So X squared by a square flus vice whereby a squared minus they expire by c square imposed. No. For this article you can see mhm Every single day what the circle you can say therefore, yeah, no X squared plus y squared equals two. I respect. So this will give us our square by he squared minus that square. Right? Sees what he goes to work, isn't it? Yes. Mm. Yeah. Yeah. Yeah. So we can say at z equals to zero Ed That equals to zero. The diameter is minimum. Yeah. Which is also equal to 200 m. Or you can see our equals two 100 100 m by because the radius is half of the diameter. Right? So now we can write instead of our the value The values 100 square by you can say is here under squared by a minus zero by C. Square. Thank you. This is there is a zero here so this is equal to one. So this really give us yeah It was 200 years and drink makers. Okay. So we can say mhm Okay. No egg. Okay Z equals two 500 m by because 500 m above the base we are given with the value of radios here. So we can say Ed Their request of five 100 million. Sorry 500 m. The diameter at base it was too to 80 m which is equal to r equals to 1 40 half of it 1/2 of the diameter. So this will give us 1 40 by 100. Whole squared minus 500 by see Whole Square. It was 2 1. So this will give us 500. Bye. See whole square It was 2140 by 100 was led by next one. So that's gonna give us one point 9 6 -1. Just give us .96. Now the value becomes one by C squared equals two .96 divided by 500 spot. So this will be 96 divided by 25. Yeah. Thanks. Mm. Yeah. And these two zeros are more included 40.96. Right? So .96 if we remove this uh Testament place then we will add two more zeros. So this will give us six by 15 6 25. Mhm. Now we can easily say hyper Berlin question virility. We know eggs, we know A. We know B. And we know see it. But this movie what? It is good by 10,000 guys. Why square by then 1007. And this this will be -6. That's where divided by 15 6 25. This will be close to one. So this is a hyper boy Lloyd equation or this. I hope you understood difference it. Thank you for watching.