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.
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.