Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

From a point \(P\) on level ground, the angle of elevation of the top of a tower is \(26^{\circ} 50^{\prime}\). From a point \(25.0\) meters closer to the tower and on the same line with \(P\) and the base of the tower, the angle of elevation of the top is \(53^{\circ} 30^{\prime}\). Approximate the height of the tower.

Step-by-Step Solution

Answer

The height of the tower is approximately 20.32 meters.

Step 1: Understand the Problem

We have two known points, point \(P\) and another point 25 meters closer to the tower. For each, we know the angles of elevation to the top of the tower. We need to find the height of the tower using trigonometric principles.

Step 2: Assign Variables

Let's denote the height of the tower as \(h\), and the distance from point \(P\) to the base of the tower as \(d\). The point 25m closer will therefore be at \(d - 25\) meters from the base of the tower.

Step 3: Use Trigonometry at Point P

Using the angle \(26^{\circ} 50^{\prime}\), the tangent relationship \(\tan(26^{\circ} 50^{\prime}) = \frac{h}{d}\) gives us the equation: \[ h = d \cdot \tan(26^{\circ} 50^{\prime}) \]

Step 4: Use Trigonometry at the Closer Point

Using the angle \(53^{\circ} 30^{\prime}\), the tangent relationship \(\tan(53^{\circ} 30^{\prime}) = \frac{h}{d - 25}\) gives us another equation: \[ h = (d - 25) \cdot \tan(53^{\circ} 30^{\prime}) \]

Step 5: Equate the Heights

Since \(h\) is the same in both equations, we can equate them: \[ d \cdot \tan(26^{\circ} 50^{\prime}) = (d - 25) \cdot \tan(53^{\circ} 30^{\prime}) \] Solve this equation for \(d\).

Step 6: Solve for d

Rearrange the equation: \[ d \cdot 0.5040 = (d - 25) \cdot 1.3270 \] Now calculate: \[ 0.5040d = 1.3270d - 33.175 \] \[ 33.175 = 1.3270d - 0.5040d \] \[ 33.175 = 0.8230d \] \[ d = \frac{33.175}{0.8230} \approx 40.31 \text{ meters} \]

Step 7: Calculate the Height of the Tower

Substitute \(d\) back into either heights equation, for instance, \[ h = 40.31 \cdot 0.5040 \approx 20.32 \text{ meters} \]This gives us the height of the tower.

Key Concepts

Angle of Elevation

The angle of elevation is a crucial concept in trigonometry, especially in problems involving heights and distances. Imagine you are looking at the peak of a tower or a mountain from a certain point on the ground. The angle between your line of sight and the horizontal line is called "the angle of elevation."
This angle is always measured from the horizontal upward, making it a positive angle. In practice, when you know the angle of elevation and one other distance (usually horizontal), you can calculate heights using trigonometric functions. This is a common method to determine the height of objects like towers or trees.
In our problem, we were given two angles of elevation from two different points. Knowing these angles allows us to set up trigonometric equations that help in calculating the object's height.

Tangent Function

The tangent function is one of the primary trigonometric functions. It's quite useful in problems dealing with right triangles. Essentially, the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
Mathematically, this is expressed as: \[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
In the context of our height and distance problem, the tangent function helps us relate the height of the tower (the opposite side) to the horizontal distance from the point of observation (the adjacent side).
Using the given angles of elevation and distances, we apply the tangent function to find missing lengths and essentially solve the problem. It's such a handy function because when elevation angles and distances are known, the height can be easily calculated using tangent.

Trigonometric Problem Solving

Trigonometric problem solving involves using the relationships between angles and sides of triangles to find unknown distances or measurements. Problems like the one we solved often require a solid understanding of trigonometric functions, such as sine, cosine, and particularly tangent for right triangles.
The typical approach involves:

• Identifying the given values and what you need to find.
• Drawing a diagram to visualize the scenario and simplify the problem.
• Applying the appropriate trigonometric function to set up equations based on the problem's data.
• Solving these equations to find the unknown values, like distances or heights.

In this exercise, we integrated all these steps by using tangent ratios for different angles of elevation to ultimately find the height of the tower.

Heights and Distances

Heights and distances is an application area of trigonometry in real-world scenarios. It involves calculating the distance to an object or the height of an object when viewed from a particular point.
Such calculations are widely used in fields ranging from architecture and engineering to navigation and landscaping. The problem we discussed reflects a classic application where you need to figure out a vertical height using horizontal distances and angles of elevation.
To solve these problems:

• Start by identifying the relation between the height to be found and the distance given.
• Use trigonometric functions like tangent to relate these values through known angles of elevation.
• Formulate and solve equations based on these relationships.

These applications show how practical and immensely useful trigonometry is in solving everyday problems that involve measuring inaccessible heights.