A surveyor is standing 100 meters away from the base of a building. Using a theodolite, the surveyor measures the angle of elevation to the top of the building to be 30 degrees. Calculate the height of the building.

3.8 The Tangent Function - Review

Explain how the period of the tangent function is determined and calculate the period of the function $y = \tan (2 x)$ .

Describe the asymptotes of the tangent function and determine the vertical asymptotes of $y = \tan (x - \frac{π}{4})$ .

Identify the key points of the function $y = \tan (x)$ within one period and explain their significance.

How does the transformation $y = \tan (x - \frac{π}{3})$ affect the graph of the tangent function?

Given the function $y = \tan (\frac{x}{2})$ , determine the period and describe how the graph compares to the standard tangent function.

Explain how the periodicity of the tangent function is determined and describe its significance in graphing the function.

Describe the behavior of the tangent function near its vertical asymptotes and explain why these asymptotes occur.

Sketch the graph of the tangent function over one period and label the vertical asymptotes and intercepts.

How does the graph of the tangent function change when it is transformed by a vertical stretch factor of 2?

10.

Explain how the tangent function can be used to model real-world phenomena, providing an example.

11.

A surveyor is standing 100 meters away from the base of a building. Using a theodolite, the surveyor measures the angle of elevation to the top of the building to be 30 degrees. Calculate the height of the building.

12.

A person is looking at the top of a lighthouse from a boat. The angle of depression from the top of the lighthouse to the boat is 15 degrees. If the lighthouse is 50 meters tall, how far is the boat from the base of the lighthouse?

13.

A pilot is descending towards an airport runway. The angle of depression from the plane to the runway is 5 degrees, and the plane is currently at an altitude of 2000 meters. How far is the plane from the runway horizontally?

14.

A cable car is traveling up a mountain. The angle of elevation from the base station to the cable car is 25 degrees, and the horizontal distance from the base station to the cable car is 300 meters. Calculate the vertical height of the cable car above the base station.

15.

A person is standing on a cliff and looking down at a boat in the water. The angle of depression to the boat is 20 degrees, and the cliff is 80 meters high. How far is the boat from the base of the cliff horizontally?

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Instructor solution

AI InstructorAUG 12, 2024, 11:47:30 AM

To find the height of the building, we can use the tangent function. The tangent of the angle of elevation is equal to the opposite side (height of the building) divided by the adjacent side (distance from the building).

$\tan (30^{\circ}) = \frac{height}{100}$

$height = 100 \times \tan (30^{\circ})$

$height = 100 \times \frac{1}{\sqrt{3}} \approx 57.74 meters$

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If the surveyor moves 50 meters closer to the building, what will be the new angle of elevation to the top of the building?

Select one of the following options:

A.
60 degrees
B.
30 degrees
C.
Approximately 45 degrees
D.
90 degrees

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