Problem 34 - Trigonometry

Question

Angle of Elevation A 155 -foot antenna is on top of a tall building. From a point on the ground, the angle of elevation to the top of the antenna is \(28.5^{\circ}\), while the angle of elevation to the bottom of the antenna from the same point is \(23.5^{\circ}\). How tall is the building?

Step-by-Step Solution

Answer

The height of the building is approximately 370.9 feet.

Step 1: Understanding the Problem

We need to determine the height of the building on which a 155-foot antenna is mounted. From a specific point on the ground, we are given two angles of elevation: one to the top of the antenna (28.5°) and another to the base of the antenna (23.5°).

Step 2: Set Up the Situation

Let the height of the building be \( h \). We denote the horizontal distance from the building to the point of observation on the ground as \( d \). The total height from ground to the top of the antenna is \( h + 155 \) feet.

Step 3: Apply Trigonometry for Top of Antenna

Using the angle of elevation to the top of the antenna (28.5°), the tangent function is used: \[ \tan 28.5^{\circ} = \frac{h + 155}{d} \] Thus, \[ d \cdot \tan 28.5^{\circ} = h + 155 \]

Step 4: Apply Trigonometry for Bottom of Antenna

Using the angle of elevation to the base of the antenna (23.5°), we apply the tangent function: \[ \tan 23.5^{\circ} = \frac{h}{d} \] Thus, \[ d \cdot \tan 23.5^{\circ} = h \]

Step 5: Solving the Equations

From the equations derived:1. \( d \cdot \tan 28.5^{\circ} = h + 155 \),2. \( d \cdot \tan 23.5^{\circ} = h \)Subtract the second equation from the first to eliminate \( h \):\( d \cdot \tan 28.5^{\circ} - d \cdot \tan 23.5^{\circ} = 155 \)Factoring out \( d \):\( d \cdot (\tan 28.5^{\circ} - \tan 23.5^{\circ}) = 155 \) Solve for \( d \):\[ d = \frac{155}{\tan 28.5^{\circ} - \tan 23.5^{\circ}} \]

Step 6: Calculate Building Height

With \( d \) known, substitute back into \( d \cdot \tan 23.5^{\circ} = h \) to find \( h \):\[ h = \frac{155 \cdot \tan 23.5^{\circ}}{\tan 28.5^{\circ} - \tan 23.5^{\circ}} \] Calculate this using a calculator.

Key Concepts

Angle of Elevation

The angle of elevation is a common term in trigonometry, and it refers to the angle between the horizontal ground and the line of sight from an observer to a point above the horizontal level they are observing. Imagine you are standing on the ground and looking up at the top of a tall building or structure. The line of sight from your eyes to the top forms an angle with the flat ground.

In our exercise example, there are two angles of elevation given:

• The angle to the top of a 155-foot antenna is 28.5 degrees.
• The angle to the bottom of the antenna is 23.5 degrees.

These angles are crucial, as they determine how we set up our calculations to find unknown heights. Visualizing these angles helps us use right triangle relationships to solve problems in trigonometry. You can associate these concepts with everyday scenarios like watching a kite soar or observing a plane take off.

Tangent Function

The tangent function is a fundamental aspect of trigonometry and is incredibly useful in solving angle of elevation problems. In a right triangle, the tangent of an angle represents the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, this is expressed as:

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

For the exercise,

• The top angle of elevation's tangent is: \( \tan 28.5^{\circ} = \frac{h + 155}{d} \)
• The bottom angle's tangent is: \( \tan 23.5^{\circ} = \frac{h}{d} \)

Here, \( h \) is the height of the building and \( d \) is the horizontal distance to the point of observation. This function allows us to formulate equations that express the relationship between distances and angles.

Solving these equations helps us find unknown parameters like distances or heights, which in turn, provides a practical understanding of the tangent function in real-world contexts.

Height Calculation

Calculating the height of a structure using angles of elevation involves applying trigonometric functions and solving equations. In this problem, we were tasked with finding the height of a building when given angles of elevation to an antenna's top and base.

Using the tangent function equations,

• We know: \( d \cdot \tan 28.5^{\circ} = h + 155 \)
• And: \( d \cdot \tan 23.5^{\circ} = h \)

By subtracting the second equation from the first, we eliminate \( h \), obtaining:

• \( d \cdot (\tan 28.5^{\circ} - \tan 23.5^{\circ}) = 155 \)

This gives us the horizontal distance \( d \). With \( d \) known, substitute back into one equation:

• \( h = \frac{155 \cdot \tan 23.5^{\circ}}{\tan 28.5^{\circ} - \tan 23.5^{\circ}} \)

This gives the exact height of the building. Such calculations highlight the power of trigonometry to solve practical problems, helping us understand the world around us better through mathematics.