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Chapter 10: Problem 64

Sighting the top of a building, a surveyor measured the angle of elevation to be \(22^{\circ}\). The transit is 5 feet above the ground and 300 feet from the building. Find the building's height. Round to the nearest foot.

Short Answer

Expert verified
The building's height is approximately 124 feet.

Step by step solution

01

Determine the angles and sides

We draw a right-angled triangle with the given elements: the base being the distance between the transit and the building (300 feet), the angle of elevation \(22^{\circ}\), and the right angle between the base and the building. The hypotenuse is the direct line of sight from the transit to the top of the building and the height is the building height subtracted by the transit's height (if the building is taller than 5 feet).
02

Apply the trigonometric ratio

We use the tangent (tan) ratio which in a right-angle triangle is defined as opposite/adjacent side. In our case, the opposite side is the building's height minus the transit's height (if the building is taller) and the adjacent side is 300 feet. Thus, we have the equation: \(tan(22^{\circ}) = (Building's Height - 5) / 300 \)
03

Solve for the building's height

We multiply both sides of the equation by 300 to isolate the building height. We get: \(Building's Height - 5 = 300 \cdot tan(22^{\circ})\). We then write the final equation for the building's height: \(Building's Height = 300 \cdot tan(22^{\circ}) + 5\).
04

Calculate relevant values

Finally, we enter the given values into the final equation and calculate the result: \(Building's Height = 300 \cdot tan(22^{\circ}) + 5 \approx 124 \) feet. We round to the nearest foot.

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