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

a). The equation of hyperboloid has the form:

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

Since crossing sections are circular, we receive that $a=b$ . We rewrite the equation in

the following form:

$x^2+y^2=a^2+\frac{a^2}{c^2}z^2$

We assume that the minimal cross-sectional radius is $10m$ . It means that $a=10m$. Using that the maximal radius is $15m$ and the height is $40m$ , we find coefficient $c$ from the equality: $100+\frac{100}{c^2}\cdot1600=225\,\, \Longrightarrow\,\,c^2=1280\,\, \Longrightarrow c=16\sqrt{5}$

Thus, the equation has the form:

$\frac{x^2}{5}+\frac{y^2}{5}-\frac{z^2}{64}=20$

b). We see that $\frac{x^2}{a^2}+\frac{y^2}{b^2}=(cosh(v))^2.$ Using hyperbolic identities, we get $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$

c). The main reason is: hyperbolic cylinder does not have circular sections and therefore does not look like a tower.