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

Hello,

as  the bases are circles,
equation is :

$\dfrac{x^2}{a^2} + \dfrac{y^2}{a^2}- \dfrac{z^2}{c^2}=1\\\\ if\ x=100,y=0,z=0\ then \ \dfrac{100^2}{a^2}=1\\\\ a=100\\\\ if\ x=150,y=0,z=-500\ then \\ \dfrac{150^2}{100^2} + \dfrac{0^2}{100^2}- \dfrac{500^2}{c^2}=1\\\\ c=100*\sqrt{2} \\\\ \boxed{\dfrac{x^2}{100^2} + \dfrac{y^2}{100^2}- \dfrac{z^2}{2*100^2}=1}$

Answer 2

Okay so the equation you need to use is (x2/a2)+(y2/b2)-(z2/c2)=1

So 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 would be:

x^2 / 100^2 + y^2 / 100^2 – z^2 / 2 * 100^2 = 1

I am hoping that this answer has satisfied your query and it will be able to help you in your endeavor, and if you would like, feel free to ask another question.