Problem 88 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. A surveyor stands 180 feet from the base of a building. The angle of elevation of the top of the building with respect to the location of the surveyor is \(57^{\circ} .\) Find the height of the building.

Step-by-Step Solution

Answer

After calculating the height by using the formula \(h = 180 \cdot \tan(57^{\circ})\), rounding to four decimal places as directed in the given problem.

Step 1: Identify given quantities

Here, the distance from the building to the surveyor which can be considered as the base of the triangle is 180 feet and the angle of elevation is \(57^{\circ}\). The objective is to find the opposite side, i.e., the height of the building \(h\).

Step 2: Use the tangent formula

The formula for tangent of an angle in a right triangle is, \(\tan(\text{angle}) = \frac{\text{opposite side}}{\text{adjacent side}}\). Here, the adjacent side is the distance from the building to the surveyor, and the opposite side is the height of the building that we want to find.

Step 3: Apply the known values

By substituting the known values into the formula, we get: \(\tan(57^{\circ}) = \frac{h}{180}\). To find \(h\), we need to isolate it by multiplying both sides of the equation by 180: \(h = 180 \cdot \tan(57^{\circ})\)

Step 4: Compute the height

Now, by computing the product \(180 \cdot \tan(57^{\circ})\), the height \(h\) can be determined. Make sure to round the result to 4 decimal places.

Key Concepts

Angle of Elevation

The angle of elevation is a concept often encountered in trigonometry and real-life applications, such as construction and navigation. It refers to the angle between the horizontal line from the observer and the line of sight to an object above the observer. The term 'elevation' signifies that this is an upward angle, whereas a downward angle would be called an 'angle of depression.' In the case of the surveyor mentioned in the problem, the angle of elevation is the angle formed by a horizontal line from the surveyor's eye to the base of the building and the line of sight from the eye to the top of the building.

To visualize it, imagine standing in front of the building and looking up at the top; the angle your line of sight makes with a level (horizontal) line is the angle of elevation. Here, it is given as \(57^\circ\), which is relatively steep, indicating the building is quite tall or that the surveyor is relatively close to its base. It is crucial for accurate measurements in practical situations, like determining the building’s height, to take this angle into consideration.

Tangent Formula

The tangent formula is fundamental in right triangle trigonometry. It relates an angle with the ratio of the lengths of the opposite side to the adjacent side in a right-angled triangle. The formula is written as \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\). In our exercise, where we have a right triangle formed by the surveyor, the building, and the line from the surveyor to the top of the building, the tangent of the angle of elevation (\( \theta \)) can be used to find the unknown height of the building (opposite side) when the distance to the building (adjacent side) is known.

By rearranging the formula, we can solve for the unknown side: if \( \tan(\theta) = \frac{h}{d} \), where \( h \) is the height of the building and \( d \) is the distance from the surveyor to the base, then \( h = d \cdot \tan(\theta) \). This formula is immensely powerful because it provides a method to calculate heights or distances that would otherwise be difficult to measure directly.

Trigonometric Applications

Trigonometry is not just a set of mathematical rules but is deeply integrated into various real-world applications. Its power lies in the ability to model and solve problems involving distances, heights, and angles, which are very common in fields such as astronomy, engineering, physics, and even art. In the context of the surveyor’s problem, trigonometry is used for indirect measurement.

Surveying often involves measuring angles and distances to determine the position of points or physical details on the earth's surface. By using trigonometric functions like the tangent formula, surveyors can calculate unknown lengths or angles without physically measuring them, which is often impractical for large distances or inaccessible points. For instance, finding the height of a tall building without ascending it, or determining the distance across a body of water without crossing it, are practical problems trigonometry can solve. This exercise reflects just a snippet of how essential trigonometry is in professional practices and how the mathematical concepts students learn have real-world implications and uses.