Question

A drum used for percussion has sides modeled by a hyperbola. The total height of the drum is 27 inches. The diameter at the top of the drum is 13 inches, and the minimum diameter is 6 inches, occurring at a height of 11 inches.

Determine the equation of a hyperbola that models the sides of the drum assuming that the center of the hyperbola occurs at the height where the diameter is the least.

Answer 1

The equation of the hyperbola that models the sides of the drum is ((y-11)^2/256) - ((x^2)/49) = 1.

Step-by-step explanation:

To determine the equation of the hyperbola that models the sides of the drum, we'll use the given information.

We are told that the total height of the drum is 27 inches, the diameter at the top is 13 inches, and the minimum diameter is 6 inches at a height of 11 inches.

We can use the standard form equation for a hyperbola with a vertical transverse axis to find the equation: ((y-k)^2/a^2) - ((x-h)^2/b^2) = 1.

1. Find the center of the hyperbola. Since the center occurs at the height where the diameter is the least, the y-coordinate of the center is 11. The x-coordinate of the center is 0 because the drum is symmetrical about the y-axis.

2. Find the transverse axis length. The transverse axis is the distance between the vertices of the hyperbola, which is the difference between the maximum and minimum heights. In this case, the transverse axis length is:

   27 - 11 = 16 inches.

3. Find the conjugate axis length. The conjugate axis length is the distance between the co-vertices of the hyperbola. In this case, the difference between the top and minimum diameters is :

   13 - 6 = 7 inches.

4. Plug the values into the equation: ((y-11)^2/16^2) - ((x-0)^2/7^2) = 1.

Therefore, the equation of the hyperbola that models the sides of the drum is ((y-11)^2/256) - ((x^2)/49) = 1.