Question

A surveyor, 31 m from a building, uses a transit to measure the angle of elevation to the top of the building. The angle of elevation is 37". The transit is set at a height of 1.5 m.

a) Calculate the distance from the transit to the top of the building.

b) Calculate the height of the building.

Answer 1

To calculate the distance from the transit to the top of the building, we can use trigonometry. The tangent of the angle of elevation is equal to the height of the building divided by the distance from the transit to the building. Using this relationship, we can find that the distance is approximately 2.39 meters. To calculate the height of the building, we can use the same trigonometric relationship and find that the height is approximately 1.74 meters.

Step-by-step explanation:

In order to calculate the distance from the transit to the top of the building, we can use trigonometry. The tangent of the angle of elevation is equal to the height of the building divided by the distance from the transit to the building. Let's denote the distance from the transit to the top of the building as x. Using the given information, we can set up the equation:

tan(37°) = height of the building / x

We know that the height of the building is 1.5 m, so we can substitute that into the equation:

tan(37°) = 1.5 m / x

Solving for x, we get:

x = 1.5 m / tan(37°) = 2.39 m

Therefore, the distance from the transit to the top of the building is approximately 2.39 meters.

To calculate the height of the building, we can use the same trigonometric relationship, but this time solve for the height. Rearranging the equation, we have:

height of the building = x * tan(37°)

Substituting in the value of x we found earlier, we get:

height of the building = 2.39 m * tan(37°) = 1.74 m

Therefore, the height of the building is approximately 1.74 meters.

Answer 2

a) 38.82 m

b) 24.86 m

Step-by-step explanation:

The relevant trig relations are ...

Cos = Adjacent/Hypotenuse

Tan = Opposite/Adjacent

We assume you intend the angle of elevation to be 37°, not 37". (The latter would make the building height be 1.506 m.)

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a)

The distance from the transit to the top of the building is the hypotenuse of the right triangle whose legs are the distance to the building and the height of the building above the transit. Using the cosine relation, we have ...

Hypotenuse = Adjacent/Cos

distance from transit to building top = (31 m)/cos(37°) ≈ 38.82 m

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b)

The height of the building above the transit can be found using the tangent relation.

additional height = (31 m)tan(37°) ≈ 23.36 m

So, the height of the building is about ...

23.36 m + 1.5 m = 24.86 m