Problem 45 - Precalculus: Functions and Graphs

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 24.94 meters.

Step 1: Understand the Problem

We have two points, \(P\) and \(Q\), on the ground. Point \(Q\) is 25 meters closer to the tower than \(P\). The angles of elevation from \(P\) and \(Q\) to the top of the tower are \(26^{\circ} 50^{\prime}\) and \(53^{\circ} 30^{\prime}\), respectively. We need to find the height of the tower.

Step 2: Set Up the Geometry

Let the height of the tower be \(h\) meters. Let the distance from \(P\) to the base of the tower be \(x\) meters. The distance from \(Q\) to the base of the tower is therefore \(x - 25\) meters.

Step 3: Apply Trigonometry at Point P

Using the tangent of the angle of elevation from \(P\), we have: \(\tan(26^{\circ} 50^{\prime}) = \frac{h}{x}\). This can be rewritten as: \(h = x \times \tan(26^{\circ} 50^{\prime})\).

Step 4: Apply Trigonometry at Point Q

Using the tangent of the angle of elevation from \(Q\), we have: \(\tan(53^{\circ} 30^{\prime}) = \frac{h}{x - 25}\). This can be rewritten as: \(h = (x - 25) \times \tan(53^{\circ} 30^{\prime})\).

Step 5: Set the Equations Equal

Since both expressions in Steps 3 and 4 equal \(h\), we set them equal to each other: \(x \times \tan(26^{\circ} 50^{\prime}) = (x - 25) \times \tan(53^{\circ} 30^{\prime})\).

Step 6: Solve for x

Solve the equation from Step 5 for \(x\):1. Expand: \(x \times \tan(26^{\circ} 50^{\prime}) = x \times \tan(53^{\circ} 30^{\prime}) - 25 \times \tan(53^{\circ} 30^{\prime})\).2. Rearrange terms to isolate \(x\): \(x(\tan(26^{\circ} 50^{\prime}) - \tan(53^{\circ} 30^{\prime})) = -25 \tan(53^{\circ} 30^{\prime})\).3. Solve for \(x\): \(x = \frac{-25 \times \tan(53^{\circ} 30^{\prime})}{\tan(26^{\circ} 50^{\prime}) - \tan(53^{\circ} 30^{\prime})}\).4. Calculate \(x\) using a calculator.

Step 7: Calculate the Height of the Tower

Once \(x\) is found, use the equation \(h = x \times \tan(26^{\circ} 50^{\prime})\) to find \(h\). Substitute the calculated value of \(x\) into this equation and compute \(h\).

Key Concepts

Understanding Angle of Elevation

In trigonometry, the "angle of elevation" is an essential concept used to determine heights and distances of objects seen from a particular point. It is defined as the angle between the horizontal and the line of sight extending from the observer to the object above the horizontal line. Suppose you are standing on flat ground looking up at the top of a tree; the angle your line of sight makes with an imaginary horizontal line from your eyes is the angle of elevation.

The higher the object or the closer you are to it, the greater this angle will be.

• The angle is measured upwards from the horizontal plane.
• It is typically used in problems where you need to calculate the height of towers, buildings, or trees.
• Tools like a protractor, clinometer, or the tangent function are often used to measure or calculate it.

In the problem, two angles of elevation are given from different points to determine the tower's height.

Exploring the Tangent Function

The tangent function in trigonometry is one of the primary functions used to relate angles to side ratios in right triangles. It is understood as the ratio of the "opposite side" to the "adjacent side" when looking at an angle in a right triangle. This concept is expressed as:\[\tan(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}}\]
In our tower problem, the tangent function is crucial in setting up the right triangle formed by the tower, the ground, and the line of sight. Here’s why tangent is useful:

• It relates the height of the tower (opposite) to the distance from the observation point to the tower's base (adjacent).
• Tangent provides a mathematical way to solve for unknown side lengths using given angle measurements.
• It helps set up equations that directly lead to solving for the desired height or distance.

Thus, when given an angle of elevation and a distance, the tangent function allows us to calculate the height efficiently.

Problem-Solving with Trigonometry

Problem-solving in trigonometry often involves using given information to find unknown values like heights, distances, or angles.

The outlined problem is a classic example where multiple trigonometric principles apply to find the height of a tower from carious reference points. How do we solve such a problem effectively?

• Identify what information is given and what needs to be found. Here, you have two angles of elevation and a distance change to calculate the tower's height.
• Use appropriate trigonometric functions, like the tangent, based on the problem context.
• Set up equations based on the relationships defined by your chosen functions and solve for unknowns.
• Check units and convert angle measures, if necessary, for consistent calculations.
• Finally, substitute found values back into your equations to ensure the solution satisfies the original problem conditions.

Trigonometry allows us to transform angles and distances into solvable equations, facilitating solutions for practical height and distance problems like this one.