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QuestionFind the height of a stone face on a mountain, given angles of elevation of 28∘28^{\circ} and 31∘31^{\circ} from 800 feet away.

Studdy Solution

STEP 1

Assumptions1. The distance from the base of the mountain to the point of sighting is800 feet. . The angle of elevation to the bottom of the stone face is 28∘28^{\circ}.
3. The angle of elevation to the top of the stone face is 31∘31^{\circ}.
4. The height of the stone face is the difference between the height from the base of the mountain to the top of the stone face and the height from the base of the mountain to the bottom of the stone face.

STEP 2

We can use the tangent of the angle of elevation to find the height of the stone face. The tangent of an angle in a right triangle is the ratio of the opposite side (height in this case) to the adjacent side (distance from the base of the mountain).
Height=Distance×tan⁡(Angle of elevation)Height = Distance \times \tan(Angle\, of\, elevation)

STEP 3

First, let's find the height from the base of the mountain to the top of the stone face using the angle of elevation to the top of the face.
Heighttop=800×tan⁡(31∘)Height_{top} =800 \times \tan(31^{\circ})

STEP 4

Now, let's find the height from the base of the mountain to the bottom of the stone face using the angle of elevation to the bottom of the face.
Heightbottom=800×tan⁡(28∘)Height_{bottom} =800 \times \tan(28^{\circ})

STEP 5

The height of the stone face is the difference between the height from the base of the mountain to the top of the stone face and the height from the base of the mountain to the bottom of the stone face.
Heightface=Heighttop−HeightbottomHeight_{face} = Height_{top} - Height_{bottom}

STEP 6

Substitute the values of HeighttopHeight_{top} and HeightbottomHeight_{bottom} from the previous steps.
Heightface=800×tan⁡(31∘)−800×tan⁡(28∘)Height_{face} =800 \times \tan(31^{\circ}) -800 \times \tan(28^{\circ})

STEP 7

Calculate the height of the stone face.
After calculating the above expression, we find that the height of the stone face is approximately41.57 feet.
The height of the stone face is approximately41.57 feet.

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