Problem 33 - Trigonometry

Question

Angle of Elevation A woman entering an outside glass elevator on the ground floor of a hotel glances up to the top of the building across the street and notices that the angle of elevation is \(48^{\circ}\). She rides the elevator up three floors ( 60 feet) and finds that the angle of elevation to the top of the building across the street is \(32^{\circ}\). How tall is the building across the street? (Round to the nearest foot.)

Step-by-Step Solution

Answer

The building is approximately 156 feet tall.

Step 1: Sketch the Situation

Draw a diagram to illustrate the situation. Mark the ground level of the hotel and the top of the building across the street. Indicate the angles of elevation and the height the woman ascended in the elevator.

Step 2: Define Variables

Let the height of the building across the street be \(H\) feet and the distance from the elevator to the base of the other building be \(d\) feet. These will be the variables we need to solve for.

Step 3: Set Up Equations for Each Position

Using trigonometry, set up an equation from the initial position on the ground floor. Use \(\tan(48^{\circ}) = \frac{H}{d}\). When she is 60 feet higher, the equation changes to \(\tan(32^{\circ}) = \frac{H - 60}{d}\).

Step 4: Solve for Distance to the Building (d)

From the first equation, solve for \(d\): \(d = \frac{H}{\tan(48^{\circ})}\). Substitute this expression for \(d\) in the second equation to find a relationship solely in terms of \(H\).

Step 5: Substitute and Simplify

Substitute \(d = \frac{H}{\tan(48^{\circ})}\) into the second equation: \(\tan(32^{\circ}) = \frac{H - 60}{\frac{H}{\tan(48^{\circ})}}\). Simplify this equation: \(\tan(32^{\circ}) \cdot \frac{H}{\tan(48^{\circ})} = H - 60\).

Step 6: Solve for the Height (H)

Rearrange the equation from Step 5 and solve for \(H\). This equation simplifies to \(H(\tan(32^{\circ}) + 60\cdot\tan(32^{\circ}) = H\cdot \tan(48^{\circ})\). Solve it numerically to find \(H = 156\) feet.

Key Concepts

Trigonometric Equations

Trigonometric equations often come into play when dealing with real-world problems involving angles and distances. In this context, these equations help us find unknown measurements such as heights or lengths that aren't directly measurable. Specifically, an angle of elevation problem like the one described uses these equations to connect angles observed with the actual height of a building or an object.

In the exercise mentioned, two main trigonometric equations were set up based on the angle of elevation at two different heights. The tangent of an angle in trigonometry is defined as the ratio of the opposite side over the adjacent side in a right triangle. Here, that means:

• The first equation relates the initial angle of elevation, 48 degrees, to the height of the building. This sets up an equation as: \( an(48^{\circ}) = \frac{H}{d}\), where \(H\) is the height of the building, and \(d\) is the distance from the point of observation to the building’s base.
• The second equation relates the new angle of elevation, 32 degrees, seen after rising 60 feet. This gives: \(\tan(32^{\circ}) = \frac{H - 60}{d}\).

Understanding these equations is critical because they form the foundation of the problem-solving process in trigonometry.

Tangent Function

The tangent function is a crucial concept in trigonometry, especially in problems involving angles of elevation and depression. This function connects the angle in a right triangle to the ratio of its opposite side to its adjacent side. In essence, the tangent of an angle \(\theta\) is given by:

• \(\tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\)

In the original problem, the tangent function was used to translate the angle of elevation into a ratio that could be used to relate the height of the building to the horizontal distance from the observer.

• Initially at ground level, the angle of 48° gave the equation \(\tan(48^{\circ}) = \frac{H}{d}\).
• After rising 60 feet, the angle changed, leading to \(\tan(32^{\circ}) = \frac{H - 60}{d}\).

By understanding how to manipulate and solve these tangent-based equations, we can determine the relationship between the height and the distance directly.

Problem Solving in Trigonometry

Problem solving in trigonometry often involves breaking down a real-life scenario into an understandable mathematical model. This is effectively achieved using sketches, assigning variables, setting up equations, and solving those equations. In our angle of elevation problem:

The process started with creating a visual representation of the scenario. This involves drawing the buildings, labeling the angles, and assigning variables for unknowns (like the height \(H\) and the horizontal distance \(d\)). Sketching transforms abstract problems into comprehensible forms.

The next step was translating these visual cues into trigonometric equations using the tangent function. We set up equations based on the woman’s position relative to the building.

• This approach breaks a complex problem into smaller steps: identify components, relate them using trigonometric functions, and solve each equation successively.
• By substituting values and rearranging equations, you can finally solve for the unknowns, making the invisible, visible.

Solving this kind of problem not only uses trigonometry practically but also hones analytical skills by applying theoretical concepts to practical situations.