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.

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.

_____

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.

lhabdulsamirahmed lhabdulsamirahmed · Answer 2 · 2022-03-03T12:17:57+01:00

The distance Pete must climb to reach the monument is 19.56 foot

Data;

Angle of elevation = 50 degrees
Angle pf depression = 22 degrees
height of the monument = 40 foot

Sine Rule

The sine rule is used to calculate the distance between a point in a scenario where e have the value of 2 angles and one side.

[tex]\frac{sinA}{a} = \frac{sinB}{b}[/tex]

Let's substitute the values for this and find the distance

[tex]\frac{sinA}{a} = \frac{sinB}{b} \\\frac{sin 50}{40} = \frac{sin 22}{b} \\ b = \frac{40*sin22}{sin50}\\ b = 19.56 foot[/tex]

The distance Pete must climb to reach the monument is 19.56 foot

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