To find the height of the building, we can use trigonometry. The scenario forms a right triangle where:

• The distance from the surveyor to the base of the building is the adjacent side of the triangle, which is 300 ft.
• The angle of elevation is 24 degrees.
• The height of the building is the opposite side of the triangle, which we will call $h$.
• The height of the surveyor's device is 5 ft above the ground, so we need to subtract this height from the total building height.

Step 1: Use the tangent function

The tangent of an angle in a right triangle is the ratio of the opposite side (height of the building) to the adjacent side (distance from the surveyor to the base of the building). So:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

In this case, we have:

$\tan(24^\circ) = \frac{h}{300}$

Step 2: Solve for $h$

We can now solve for the height of the building (excluding the surveyor's device height):

$h = 300 \cdot \tan(24^\circ)$

Step 3: Adjust for the surveyor's height

The total height of the building is:

$\text{Total height of the building} = h + 5$

Let’s calculate the value for $h$ and the total height of the building.

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I'll now compute the result. math

Given data

distance = 300 # distance from the surveyor to the base of the building in feet angle_deg = 24 # angle of elevation in degrees surveyors_height = 5 # height of the surveyor's device in feet

Convert angle to radians for math.tan function

angle_rad = math.radians(angle_deg)

Calculate the height of the building (excluding surveyor's height)

height_building = distance * math.tan(angle_rad)

Total height of the building including surveyor's device height

total_height = height_building + surveyors_height

Round the result to the nearest tenth

round(total_height, 1)The total height of the building, including the 5 ft height of the surveyor's device, is approximately 138.6 feet.

Would you like further details or clarification on any part of this process?

Here are 5 related questions you may find helpful:

1. How does the tangent function work in solving real-world problems like this one?
2. What other trigonometric functions can be used to solve for heights and distances?
3. How can I calculate the height of a building if I know the distance and angle but not the height of the surveyor's device?
4. What happens if the surveyor is positioned at a different distance from the building? How would that affect the height calculation?
5. How do you convert angles from degrees to radians, and why is it necessary in trigonometric calculations?

Tip: Always ensure your calculator is set to the correct angle mode (degrees or radians) depending on the problem you're solving.