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As shown in the figure below, a cable car carries passengers from a point A, which is 1.2 miles from a point B at the base of a mountain, to a point P at the top of the mountain. The angles of elevation of P from A and B are 𝛼 = 24° and 𝛽 = 65°, respectively. (Round your answers to one decimal place.)

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  • 2 months ago
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    Note that given 𝛽, we can calculate the measures of ∠ABP and ∠APB as follows:

    ∠ABP = 180 -  𝛽 = 115

    ∠APB = 𝛽 - 𝛼  = 41

    Calculating these angles allows us to use the Law of Sines as follows:

      [AP / sin(115) ]= [ 1.2 / sin(41) ]

    Solving for AP gives

    AP = 1.2[ sin(115) / sin(41) ] ≈ 1.7m

    Thus, the distance between A and P is approximately 1.7 miles.

    Also note we can drop a perpendicular of height h from point P to the ground, forming a right triangle. This perpendicular is in fact the height of the mountain. Thus, from trig ratios we have

    sin(𝛼) = h / 1.7

    Solving for h gives

    h = 1.7sin(𝛼) ≈ 0.7 miles

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