Mathematics

Question

Can someone help me
Can someone help me

2 Answer

  • Answer:

       6 < x < 23.206

    Step-by-step explanation:

    To properly answer this question, we need to make the assumption that angle DAC is non-negative and that angle BCA is acute.

    The maximum value of the angle DAC can be shown to occur when points B, C, and D are on a circle centered at A*. When that is the case, the sine of half of angle DAC is equal to 16/22 times the sine of half of angle BAC. That is, ...

      (2x -12)/2 = arcsin(16/22×sin(24°))

      x ≈ 23.206°

    Of course, the minimum value of angle DAC is 0°, so the minimum value of x is ...

      2x -12 = 0

      x -6 = 0 . . . . . divide by 2

      x = 6 . . . . . . . add 6

    Then the range of values of x will be ...

      6 < x < 23.206

    _____

    * One way to do this is to make use of the law of cosines:

      22² = AB² + AC² -2·AB·AC·cos(48°)

      16² = AD² + AC² -2·AD·AC·cos(2x-12)

    The trick is to maximize x while satisfying the constraints that all of the lengths are positive. This will happen when AB=AC=AD, in which case the equations be come ...

      22² = 2·AB²·(1-cos(48°))

      16² = 2·AB²·(1 -cos(2x-12))

    The value of AB drops out of the ratio of these equations, and the result for x is as above.

  • Answer 6<x<30:

    Step-by-step explanation:

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