12.2 The hyperbola  (Page 7/13)

We are assuming the center of the tower is at the origin, so we can use the standard form of a horizontal hyperbola centered at the origin: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ where the branches of the hyperbola form the sides of the cooling tower. We must find the values of $a^2$ and $b^2$ to complete the model.

First, we find $a^2.$ Recall that the length of the transverse axis of a hyperbola is $2a.$ This length is represented by the distance where the sides are closest, which is given as $65.3$ meters. So, $2a=60.$ Therefore, $a=30$ and $a^2=900.$

To solve for $b^2,$ we need to substitute for $x$ and $y$ in our equation using a known point. To do this, we can use the dimensions of the tower to find some point $\left(x,y\right)$ that lies on the hyperbola. We will use the top right corner of the tower to represent that point. Since the y -axis bisects the tower, our x -value can be represented by the radius of the top, or 36 meters. The y -value is represented by the distance from the origin to the top, which is given as 79.6 meters. Therefore,

The sides of the tower can be modeled by the hyperbolic equation