Question

A 40 foot tall monument sits on top of a hill. Pete is standing at a point in the hill and observes the top of the monument at an angle elevation of 50 degrees and the bottom of the monument at an angle of elevation of 22 degrees. Find the distance Pete must climb to reach the monument.

Answer 1

Answer:

  55 feet

Step-by-step explanation:

Please refer to the attached diagram. We assume the distance of interest is the length of segment AC.

The Law of Sines can be used to find the length AC. It is opposite angle B, which is the complement of the elevation angle 50°. The known side of the triangle is AB, which is 40 feet. It is opposite ∠ACB, which is the difference between the elevation angles, an angle of 28°.

The Law of Sines tells us ...

  AC/sin(B) = AB/sin(∠ACB)

Multiplying by sin(B), we have ...

  AC = AB·sin(B)/sin(∠ACB) = 40·sin(40°)/sin(28°) ≈ 54.767 ft

Pete must climb about 55 feet to reach the base of the monument.

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Alternate interpretation

The height of the base of the monument above Pete's observation point is ...

  (54.767 ft)·sin(22°) = 20.5 ft

This is the change in elevation required for Pete to reach the monument. The wording "distance Pete must climb" is ambiguous, so we cannot tell if it is the distance Pete must move uphill along the ground, or the change in altitude.