Question

As an engineer,you have been asked to design a cooling tower in the shape of hyperboloid of one sheet. the horizontal cross sections of the cooling tower are circular with 10m. the cooling tower is 40m tall with maximum cross-sectional radius of 15m. construct a mathematical equation for this cooling tower.

Answer 1

Since horizontal radius is 10m and maximum radius is 15m. Since maximum height of the hyperboloid is 10m.

Equation of hyperboloid is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1$

Since region in horizontal surface is circular. so a = b = 10

Maximum radius will be when z is maximum but z will be 5m as half distance is up side and other half is in down side.

so equation will be

$\frac{x^{2}}{10^{2}} + \frac{y^{2}}{10^{2}} - \frac{z^{2}}{c^{2}} = 1$

putting condition that x = y = 15 when z = 5 then $c^{2} = \frac{50}{7}$

Hence required equation is

$\frac{x^{2}}{10^{2}} + \frac{y^{2}}{10^{2}} - \frac{z^{2}}{\frac{50}{7}} = 1$