What steps are involved to derive a functional expression for the revolving line of a cooling tower?

Question

I am in my second year of International Baccalaureate diploma program. I have chosen to evaluate the integral expression of the Cooling Tower numerically and analytically for my Internal Assessment from Mathematics HL. So far, I have succeeded in performing indirect measurement of a cooling tower from a photograph, transforming measurements into real dimensions using scale factor and numerically evaluating definite integral by Trapezoidal rule.

I am having difficulties in modeling two functions of hyperbolas joined in a common vertex. I have transformed a standard form into general equation of hyperbola and solved for x. However, in my case, the final result differs with the one I have found in cooling-tower handout. Thus, I would like to ask you for an explanation or at least hints how to get the equation of the right branch of the hyperbola.

My calculations

Thank you.

Answer 1

You are forgetting that $a$ and $b$ are just arbitrary constants, and falsely assuming that they must labeled by the same letters at all times. You have correctly calculated that if $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ then the right branch of the hyperbola (i.e., the one with $x > 0$) is given by $$x = \frac ab\sqrt{b^2 + y^2}$$

But let's bring that $\frac ab$ inside: $$x = \sqrt{\frac{a^2}{b^2}(b^2 + y^2)} = \sqrt{a^2 + \frac{a^2}{b^2}y^2}$$

Now let's define $c = \frac {a^2}{b^2}$. Then we have $$x = \sqrt{a^2 + cy^2}$$ which is identical to the equation in the handout, except for using $c$ instead of $b$. And that is your answer. The value you labeled $b$ is not the same as the value labeled $b$ in the handout.