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Magazine Chapter 5: Problem 45
Height of a tower 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.
Short Answer
Expert verified
The height of the tower is approximately 30.18 meters.
Step by step solution
01
Understand the problem
Two points are on the same line with a distance of 25.0 meters between them. The angles of elevation to the tower's top are given for each point. We need to find the tower's height.
02
Set up the problem using trigonometry
The height of the tower, denoted as \( h \), can be found using trigonometric tangent functions. Let \( x \) be the distance from point \( P \) to the base of the tower. The tangent of the angles from the two points gives us two equations: \( \tan(26^{\circ} 50') = \frac{h}{x} \) and \( \tan(53^{\circ} 30') = \frac{h}{x - 25} \).
03
Solve the first equation for x
Use the first equation \( \tan(26^{\circ} 50') = \frac{h}{x} \) to solve for \( h \) in terms of \( x \). This gives \( h = x \cdot \tan(26^{\circ} 50') \).
04
Solve the second equation for h
Use the second equation \( \tan(53^{\circ} 30') = \frac{h}{x - 25} \) to solve for \( h \) in terms of \( x - 25 \). This gives \( h = (x - 25) \cdot \tan(53^{\circ} 30') \).
05
Equate the two expressions for h
Set the expressions for \( h \) from Step 3 and Step 4 equal: \( x \cdot \tan(26^{\circ} 50') = (x - 25) \cdot \tan(53^{\circ} 30') \).
06
Solve for x
Solve the equation from Step 5 for \( x \). Substitute the tangent values, rearrange the terms and isolate \( x \).
07
Calculate x numerically
Using \( \tan(26^{\circ} 50') \approx 0.5095 \) and \( \tan(53^{\circ} 30') \approx 1.3369 \), plug these values into the equation: \( x \cdot 0.5095 = (x - 25) \cdot 1.3369 \). Simplify to find \( x \approx 59.27 \).
08
Find the height of the tower
Substitute \( x \) back into \( h = x \cdot \tan(26^{\circ} 50') \) to find \( h \). This gives \( h = 59.27 \cdot 0.5095 \approx 30.18 \).
09
Round to an appropriate measure
Round \( h \approx 30.18 \) to two decimal places as 30.18 meters.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angle of Elevation
The angle of elevation is a fundamental concept in trigonometry. It is defined as the angle formed by the horizontal line and the line of sight, looking up to an object. In our exercise, the angles of elevation to the top of a tower from two different points are provided. These angles are measured from the ground level, which assists in solving for the tower's height.
- Angle of elevation is measured from the horizontal plane upwards.
- It is crucial for calculating heights and distances in real-life situations.
- In the given problem, angles of 26掳 50鈥� and 53掳 30鈥� are used to find the tower's height.
Tangent Function
The tangent function is one of the primary trigonometric functions used to relate angles to side lengths. It is defined in a right triangle as the ratio of the opposite side to the adjacent side. In our specific problem, it helps in constructing the equations necessary for finding the tower's height.
- The formula for tangent is: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
- In the problem, \( \tan(26^{\circ} 50') = \frac{h}{x} \) and \( \tan(53^{\circ} 30') = \frac{h}{x - 25} \) are set up to find the height \( h \).
- The tangent values used are approximate: \( \tan(26^{\circ} 50') \approx 0.5095 \) and \( \tan(53^{\circ} 30') \approx 1.3369 \).
Trigonometric Equations
In trigonometry, equations are used to find unknown measures, such as sides of triangles or angles. Solving the problem at hand requires forming and solving trigonometric equations. We use known values of angles and the tangent function to set up these equations.
- We start by deriving two separate equations from our tangent function identities.
- Equate the two expressions obtained from the different viewpoints to eliminate the height \( h \).
- Solve for the other unknown, which in this case is the distance \( x \) from the initial point \( P \).
Problem Solving in Mathematics
Problem-solving is a critical skill in mathematics that involves understanding, analyzing, and deriving solutions to problems. In this exercise, we demonstrate a methodical approach using trigonometry.
- Begin by comprehending what the problem requires.
- Translate the scenario into mathematical expressions using known formulas.
- Use algebraic manipulation to solve the equations.
- Finally, verify the solution under the context of the problem to ensure accuracy.
