Problem 360 - Precalculus

Question

For the following exercises, solve for the unknown sides of the given triangle. The angle of elevation to the top a building in Baltimore is found to be 4 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

Step-by-Step Solution

Answer

The building is approximately 369 feet tall.

Step 1: Understand the Problem

We are given a right triangle formed by the building and the ground with the angle of elevation to the top of the building being 4 degrees and the distance from the observation point to the base of the building being 1 mile. We need to find the height of the building.

Step 2: Use the Tangent Ratio

In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. So, we can set up the equation \( \tan(4^\circ) = \frac{\text{height of building}}{1} \).

Step 3: Solve for the Height

Rearrange the equation to solve for the height: \( \text{height of building} = \tan(4^\circ) \times 1 \). Calculate \( \tan(4^\circ) \) using a calculator to find the height. Approximate \( \tan(4^\circ) \approx 0.0699 \).

Step 4: Calculate the Height

Substitute the tangent value: \( \text{height of building} \approx 0.0699 \times 1 \) mile. Since 1 mile is approximately 5280 feet, convert the height to feet: \( 0.0699 \times 5280 \approx 369 \text{ feet} \).

Step 5: Conclusion

The height of the building is approximately 369 feet.

Key Concepts

Understanding Right Triangles

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. The two sides that form this right angle are known as the "legs," and the side opposite the right angle is the "hypotenuse." Right triangles are fundamental in trigonometry because they serve as a basis for defining trigonometric ratios such as sine, cosine, and tangent.
This type of triangle is important because right angles are involved in many practical problems, particularly in engineering and architecture. In our exercise, the right triangle is formed by the base of the building, the height of the building, and the distance from the observation point. Understanding this triangle structure allows us to apply trigonometric ratios to solve for unknown values, such as the height of the building.

Exploring the Tangent Ratio

The tangent ratio is one of the primary trigonometric ratios used to relate the angles and sides of a right triangle. It is defined specifically for angles in a right triangle as the ratio of the length of the opposite side to the length of the adjacent side.

• Mathematically, this is expressed as: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

In our problem, the angle of elevation is given as 4 degrees, and we wish to find the height of a building. Here, the height of the building is the side opposite the angle of elevation, and the distance from the base is the adjacent side.
By applying the tangent ratio, \( \tan(4^\circ) = \frac{\text{height}}{1 \text{ mile}} \), we rearrange this to solve for height: \( \text{height} = \tan(4^\circ) \times 1 \text{ mile} \). The tangent ratio thus helps us directly calculate the unknown height using the known distance.

Understanding the Angle of Elevation

The angle of elevation is a term used in trigonometry to describe the angle between the horizontal line of sight and the line of sight directed upwards towards an object. It is a vital concept that allows us to determine heights and distances of objects that are not easily measurable.
In the given exercise, the angle of elevation is 4 degrees. This angle helps determine the relationship between the observer on the ground and the top of a building. When we use the angle of elevation in conjunction with trigonometric ratios, we can effectively solve for unknown dimensions in right triangles.

• For example, the angle of elevation allows the observer to calculate the height of the building without physically measuring it.
• It is heavily used in fields such as surveying, navigation, and architecture, where direct measurements are not feasible.

By understanding and using the angle of elevation correctly, one can solve many practical problems using trigonometry.

Converting Distances Appropriately

Distance conversion is an essential step in solving practical trigonometry problems because measurements can be in varied units. Commonly in trigonometry, we work with metrics like meters, feet, kilometers, and miles. Each unit of measure needs to be standardized to ensure calculations are correct.
In our exercise, the distance from the observer to the building's base is given as 1 mile, which we convert to feet for the final calculation. Knowing that 1 mile equals 5280 feet allows precise calculations when the height is calculated: \( \text{height in miles} = \tan(4^\circ) \); converting, we get: \( \text{height in feet} = 0.0699 \times 5280 \).
Getting accurate results depends heavily on appropriate conversions, ensuring the units are consistent across all calculations.