Anonymous asked in Science & MathematicsMathematics · 1 month ago

Precalculus Question?

Felicia measures the angle of elevation of the peak of a mountain as 24˚. Donald, who is 1200 feet closer on a straight level path, measures the angle of elevation as 44˚. How high is the mountain? Assume Felicia and Donald measure the angles of elevation from their lines of sight 6 feet above the ground.

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2 Answers

  • 1 month ago

    Let d be the distance that Donald is from the peak horizontally.

    Let d+1200 be the distance that Felicia is from the peak horizontally.

    The right triangle for Felicia has an angle of elevation of 24°, an opposite leg of b and an adjacent leg of d+1200:

    tan(24°) = b / (d + 1200)

    The right triangle for Donald has an angle of elevation of 44°, an opposite leg of b and an adjacent leg of d:

    tan(44°) = b / d

    Solve the second question in terms of d:

    d = b / tan(44°)

    Plug that into the first equation:

    tan(24°) = b / (b/tan(44°) + 1200)

    Solving for b, we get:

    b = 1200 tan(24°) tan(44°) / (tan(44°) - tan(24°))

    b ≈ 991.3 ft

    But we need to add 6 feet to that to get the height of the mountain.


    ~997.3 ft

  • 1 month ago

    Let's add a few more variables.  Let's call the distance of the base of the shorter triangle "x".  So the base of the larger triangle would then be (x + 1200).

    Then the value marked as "b" is 6 feet less than the height "h":

    b = h - 6

    And is the height of both triangles.

    Tangent in right triangles is the ratio between an angle's opposite length over the adjacent length.  we have two angles so we can create two tangent equations with the two unknowns that we can then solve:

    tan(24) = b / (1200 + x) and tan(44) = b / x

    Since we want to solve for b in order to get h, let's solve the first equation for x in terms of b so we can substitute into the other equation:

    tan(44) = b / x

    x tan(44) = b

    x = b / tan(44)

    Substitute into the other equation to get:

    tan(24) = b / (1200 + x)

    tan(24) = b / [1200 + b / tan(44)]

    Let's start with getting a common denominator in the braces so we can add the fractions:

    tan(24) = b / [1200 tan(44) / tan(44) + b / tan(44)]

    tan(24) = b / {[1200 tan(44) + b] / tan(44)}

    Turn the division of fractions into the multiplication of the reciprocal:

    tan(24) = b tan(44) / [1200 tan(44) + b]

    Now multiply both sides by that denominator and distribute the tan(24) constant value into the binomial:

    tan(24)[1200 tan(44) + b] = b tan(44)

    1200 tan(44) tan(24) + b tan(24) = b tan(44)

    We can then move the second term on the left side to the right side by subtracting it from both sides:

    1200 tan(44) tan(24) = b tan(44) - b tan(24)

    Factor the b, then divide both sides by the binomial-coefficient:

    1200 tan(44) tan(24) = [tan(44) - tan(24)] b

    [1200 tan(44) tan(24)] / [tan(44) - tan(24)] = b

    That's your exact answer.  I'll get a decimal approximation and round everything to 6 DP as I work, then round to 3DP at the end:

    [1200(0.965689)(0.445229)] / (0.965689 - 0.445229) = b

    515.943297 / 0.52046 = b

    b = 991.322

    Now we can use this to solve for h, the answer to our question:

    b = h - 6

    991.322 = h - 6

    997.322 = h

    The mountain is about 997.322 feet tall.

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