Answer to Question #121458 in Calculus for Roger

Question

As an engineer, you have been tasked to design a cooling tower int he shape of a 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. A) Construct a mathematical equation for this cooling tower. B) If x=a cos(u)cosh(v), y=b sin(u)cosh(v) and z=c sin h(v), show that (x,y,z) lies on your equation in Q1(A). C) A colleague wants to construct the cooling tower using a hyperbolic cylinder, give reasons for your result in Q1(A) as the best model for the design of cooling tower.

Answer 1

(i) Since horizontal radius is 10m and maximum radius is 15m. Since maximum height of the hyperboloid is 40m.

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 $c^{2} = \frac{40}{7}$

Hence required equation is

$\frac{x^{2}}{10^{2}} + \frac{y^{2}}{10^{2}} - \frac{z^{2}}{\frac{40}{7}} = 1$

(ii) $x = a(cosu)(cosh(v)), y = b(sinu)(cosh(v)),z=(sinh(v))$

so using these points,

we can that

$(\frac{x}{a})^{2} + (\frac{y}{b})^{2} - (\frac{z}{c})^{2} =1$

which is the equation of the hyperboloid which is similar to the above equation with $a=10, b = 10, c=\frac{40}{7}$

(iii) Since the equation obtained in (i) is the desired equation of the curve that is required to construct a water cooling tower, so result of part (i) is suited for making water cooling tower.