A drinking glass has sides following the shape of a hyperbola. The minimum diameter of the glass is 45 millimeters at a height of 83 millimeters. The glass has a total height of 180 millimeters, and the diameter at the top of the glass is 57 millimeters. Find the equation of a hyperbola that models the sides of the glass assuming that the center of the hyperbola occurs at the height where the diameter is minimized.

The answer is a = 128.95 and b = 14.75

Step-by-step explanation:

Solution

Given

let the equation of the hyperbola be denoted as x2/a2 - y2/b2 = 1 here, we will consider the foci on the x-axis. All units are considered to be in mm.

From question stated, with x and y coordinates represents height and radius respectively,

Then,

When x = 83 + a, y is = 45/2 = 22.5 and

At x = 180 + a, y is = 57/2 = 28.5

It is important to know that, the height is estimated from the focus so we a a is included to the heights.

Thus,

(83+a) 2/a2 - 22.52/b2 = 1 and (180+a) 2/a2 - 28.52/b2 = 1

By using a calculator we have, a = 128.95 and b = 14.75

Therefore a = 128.95 and b = 14.75