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Math Help - A problem solving question I just don't get..

  1. #1
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    A problem solving question I just don't get..

    A helicopter is flying due west over level ground at a constant altitude of 222m and at a constant speed. An intelligent, stationary goat, which is due west of the helicopter, takes two measurements of the angle between the ground and the helicopter. The first measurement the goat makes is 15 and the second measurement, which he makes 1 minute later, is 75. If the helicopter has not yet passed over the goat, how fast is the helicopter traveling in kilometers per hour? Find exact values in your solution without using a calculator. Clearly show and explain how you derived your solution.

    Test soon, don't understand how to do this question, any help would be great. Thanks!
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  2. #2
    Member sinewave85's Avatar
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    Quote Originally Posted by zuuberbat View Post
    A helicopter is flying due west over level ground at a constant altitude of 222m and at a constant speed. An intelligent, stationary goat, which is due west of the helicopter, takes two measurements of the angle between the ground and the helicopter. The first measurement the goat makes is 15 and the second measurement, which he makes 1 minute later, is 75. If the helicopter has not yet passed over the goat, how fast is the helicopter traveling in kilometers per hour? Find exact values in your solution without using a calculator. Clearly show and explain how you derived your solution.

    Test soon, don't understand how to do this question, any help would be great. Thanks!
    Here is a quick drawing to help demonstrate what I am talking about:



    It is not to scale, but it illustrates the quantities we are working with. Drawing out the problem is always a good idea in trigonometry.

    What you know is the angles and the altitude, which constitutes the sides opposite to the known angles. Since you know the leg opposite to the angle and need to find the legs adjacent to the angles of these two right triangles(which I have labeled D1 and D2) which trigonometric ratio do you need to use?

    D1 = \frac{222}{\tan15}

    D2= \frac{222}{\tan75}

    In order to solve these without a calculator, you are going to need to use the addition laws and know the exact values of the special angles:

    \tan75 = tan(45 + 30) = \frac{tan(45) + tan(30)}{1 - tan(45)tan(30)}

    \tan15 = tan(45 - 30) = \frac{tan(45) - tan(30)}{1 + tan(45)tan(30)}

    The last step is to find the distance in meters the helicopter traveled (subtract D2 from D1) in one minute and then convert that into kilometers per hour.
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  3. #3
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    Quote Originally Posted by zuuberbat View Post
    A helicopter is flying due west over level ground at a constant altitude of 222m and at a constant speed. An intelligent, stationary goat, which is due west of the helicopter, takes two measurements of the angle between the ground and the helicopter. The first measurement the goat makes is 15 and the second measurement, which he makes 1 minute later, is 75. If the helicopter has not yet passed over the goat, how fast is the helicopter traveling in kilometers per hour? Find exact values in your solution without using a calculator. Clearly show and explain how you derived your solution.
    make a sketch ?

    let x_1 = horizontal distance from the goat to the helo when the angle = 15 degrees

    x_2 = horizontal distance from the goat to the helo when the angle = 75 degrees

    distance traveled in 1 minute = x_1 - x_2

    speed in km/hr = 60(x_1 - x_2)

    trig relationships ...

    \tan(15) = \frac{222}{x_1}

    \tan(75) = \frac{222}{x_2}

    do it.
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