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25 April, 21:28

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.

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  1. 25 April, 21:37
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    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
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