Question

A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet. The diameter at the base is 300 m and the minimum diameter, 500 m above the base, is 200 m. Find an equation describing the shape of the tower in the coordinates where the origin is at the center of the narrowest part of the tower. In particular, use coordinates where the origin is 500 m above the ground.

Answer 1

so i first tried to figure out the center of the hyperboloid. I believe it is (0,0,500)

Answer 2

then i set up the basic hyperboloid function of \[x^2+y^2-(z-500)^2\] (i am aware that the x y and z need to be divided by something) I'm just showing my thought process

Answer 3

So then i tried playing with level sets of z.
\[x^2+y^2=10000+(k-500)^2\]

Answer 4

i got the 10000 because at the center the radius is 100

Answer 5

i then tried checking this with the 150 radius at the bottom (z or k=0)
From this is found that \[(k-500)^2\] needs to be divided by 20

Answer 6

so i'm getting something like \[x^2+y^2-\frac{ (z-500)^2 }{ 20 }=1000\]..but its not right, so i'm curious what i'm doing wrong =(

Answer 7

10000*

Answer 8

or did i completely mess that up >.>

Answer 9

@phi any suggestions?

Answer 10

@jim_thompson5910 any ideas?

Answer 11

@UnkleRhaukus help if you can please

Answer 12

@satellite73

Answer 13

@Mertsj

Answer 14

i got like\[\frac{ x^2 }{ 100^2 }+\frac{ y^2 }{ 100^2}-\frac{ (z-500)^2 }{ d }=1\]

Answer 15

how do i find d..?

Answer 16

actually i think i did this completely wrong.....should the center of this be (100,100,500)?...so lost right now

Answer 17

Let's just play like we barely know anything about 3D geometry and just build the simplest mode that works!

Projections onto the x-y plan are circles.
For z = 500, we have x^2 + y^2 = 100^2 = a*z = a*500 = 5*a*100 and a = 20
At present, we seem to have x^2 + y^2 = 20z

For z = 0, we have x^2 + y^2 = 20*0 = ?? That will never do! The model must be wrong.

Let's try x^2 + y^2 = az+b. This leads to \(500a+b = 100^{2}\) and \(0a+b = 150^{2}\), which are easily solved for a = -25 and b = 22500

Just to check, 1000(-25) + 22500 = -2500. Well, THAT will never do, either.

One more.

x^2 + y^2 = az^2 + bz + c

This leads to

\(c = 150^{2}\)

\(a\cdot 500^{2} + b\cdot 500 + c = 100^{2}\)

\(a\cdot 1000^{2} + b\cdot 1000 + c = 150^{2}\)

Again, this is easily solve for a = 1/20, b = -50, and c = 150^2

Finally, we are done when we write \(20\cdot x^2 + 20\cdot y^2 = (z-500)^{2} - 500^{2} = z\cdot (z - 1000)\)

I'm sure there's a more elegant way to go about this, but it helps to know the solution exists.

Answer 18

It looks like you have the right idea, but I would start with the full equation
http://en.wikipedia.org/wiki/Hyperboloid
from Google, as I don't memorize this stuff:
\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1 \]

Next, read the question carefully
Find an equation describing the shape of the tower in the coordinates where the origin is at the center of the narrowest part of the tower

(I try to make sure every sentence is "accounted for" when writing down the requirements)

Based on a picture of what this shape looks like, we see it will be symmetric about the z-axis (vertical axis). It will have a circular shape that flutes out as you go higher. That tells me we can just concentrate on x and z. what ever we come up for x, we can use on y

Now note the points that lie on this curve