Thread: Hyperbola question

My teacher said this was badly worded but I still want to solve it:

"Two megaphones shaped like trumpets are attached to the roof of a car, sending sound waves in both directions. These megaphonees were designed to follow a hyperbola with vertices at (0, 8) and (0, -8), and foci at (0, 10) and (0, -10) in a Cartesian plane. The total length of the megaphones is 96cm. What is the diameter of the trumpets?"

The answer is: ~71cm.

Any help would be greatly appreciated!
Thanks in advance!

Originally Posted by s3a

My teacher said this was badly worded but I still want to solve it:

"Two megaphones shaped like trumpets are attached to the roof of a car, sending sound waves in both directions. These megaphonees were designed to follow a hyperbola with vertices at (0, 8) and (0, -8), and foci at (0, 10) and (0, -10) in a Cartesian plane. The total length of the megaphones is 96cm. What is the diameter of the trumpets?"

The answer is: ~71cm.

Any help would be greatly appreciated!
Thanks in advance!

I assume those (0, 8), (0, -8), (0, 10), and (0, -10) are in centimeters also.

You can write that hyperbola as $\displaystyle \frac{y^2}{64}- \frac{x^2}{b^2}= 1$. Further, the information about the foci tells you that $\displaystyle 64+ b^2= 100$ so $\displaystyle b^2= 36$.

The equation of the hyperbola is $\displaystyle \frac{y^2}{64}- \frac{x^2}{36}= 1$. The radius of the trumpets is the y value when x= 96.