In traveling across flat land, you see a mountain directly in front of you. Its angle of elevation (to the peak) is 3.5°. After you drive 15 kilometers closer to the mountain, the angle of elevation is 9° (see the following figure).
15 km
An image contains a car on the left side, a mountain on the right side and three dashed lines.
The first dashed line starts at the mountain peak, goes down and to the left and ends at the car.
The second dashed line starts at the mountain peak, goes down and to the left and ends some distance to right of the car.
The third dashed line is vertical line and goes from the base of the mountain to the peak.
The distance from the car to where the second dashed line meets the land is 15 km.
The angle made with the land and the first dashed line is 3.5°.
The angle made with the land and the second dashed line is 9°.
Approximate the height of the mountain (in kilometers). (Round your answer to one decimal place.)

Question

Answered

Answer :

sqdancefan sqdancefan · Answer 1 · 2023-12-29T15:15:38-05:00

Answer:

1.5 km

Step-by-step explanation:

You want the height of a mountain if its angle of elevation changes from 3.5° to 9° as a car moves 15 km closer.

The tangent relation is ...

Tan = Opposite / Adjacent

If h is the height of the mountain, this tells us ...

tan(3.5°) = h/(car distance)

tan(9°) = h/(car distance - 15 km)

Solving each of these equations for 'car distance', we have ...

car distance = h/tan(3.5°)

car distance = h/tan(9°) +15 km

Equating the expressions for car distance gives ...

h/tan(3.5°) = h/tan(9°) +15 km

h(1/tan(3.5)° -1/tan(9°)) = 15 km

[tex]h=\dfrac{15\text{ km}}{\dfrac{1}{\tan(3.5^\circ)}-\dfrac{1}{\tan(9^\circ)}}\approx\boxed{1.5\text{ km}}[/tex]

The height of the mountain is about 1.5 km.

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Additional comment

That's about 2405 feet high. The car was initially about 24.4 km from the mountain, and drove to 9.4 km from the base of the mountain.