Problem 71 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. The angle of elevation of the top of a tower with respect to a certain point on the ground is \(38^{\circ}\) From a point 15 feet closer to the tower, the angle of elevation is \(42^{\circ} .\) Find the height of the tower.

Step-by-Step Solution

Answer

It is the solution to the system of equations that determines the height of the tower. The exact solution will depend on the numeric solution of the system of equations described in the step-by-step solution.

Step 1: Set up the equations

Let \(h\) be the height of the tower and \(d\) the distance from the base of the tower to the first observation point. From the problem, the distance for second observation point is 15 feet closer than the first, so it can be expressed as \(d - 15\). The two angles of elevation form the following equations using the tangent of angles: \(\tan(38^{\circ}) = h/d\) and \(\tan(42^{\circ}) = h/(d-15)\)

Step 2: Solve the system of equations

Solving the first equation for \(h\): \(h = d \cdot \tan(38^{\circ})\), and substitute this in the second equation: \(d \cdot \tan(38^{\circ}) = (d - 15) \cdot \tan(42^{\circ})\). This equation can be solved for \(d\), this will give the distance from the tower to the first observation point.

Step 3: Determine the height of the tower

Once you have determined \(d\), substitute its value into the first equation: \(h = d \cdot \tan(38^{\circ})\) to find the height \(h\) of the tower.

Key Concepts

Angle of Elevation

Understanding the concept of angle of elevation is crucial for solving problems involving right triangle trigonometry. The angle of elevation is the angle formed between the horizontal ground and a line of sight directed towards an object above the horizontal plane. This concept is commonly used in navigation, architecture, and various fields of engineering.
For instance, in the exercise, the angle of elevation helps determine the height of a tower when observed from different points on the ground. The angle increases as you move closer to the base of the object. In our example, the initial angle of elevation is measured at \(38^{\circ}\), and it becomes \(42^{\circ}\) when observed from 15 feet closer.
When dealing with angles of elevation:

• Always identify the fixed horizontal line from which the angle is measured.
• Ensure you accurately measure the angle using tools like a clinometer.
• Remember that this angle is vital in applying trigonometric functions like tangent.

This knowledge aids in assessing distances and heights, fundamental in real-world applications.

Tangent Function

The tangent function is one of the fundamental trigonometric functions, especially useful in right triangle problems. It relates the angle in a right triangle to the ratio of the opposite side over the adjacent side. This makes the tangent function particularly valuable for calculating unknown lengths of sides in right triangles.
In our exercise, the tangent function is applied to create equations based on the angles of elevation for the tower. Given an angle \( \theta \), the tangent function is expressed as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). In this scenario, the calculations involve:

• Initial equation: \( \tan(38^{\circ}) = \frac{h}{d} \)
• Second equation: \( \tan(42^{\circ}) = \frac{h}{d-15} \)

Notice how \( h \) (height) is the opposite side and \( d \) (distance from the tower) is the adjacent side. By manipulating these equations, you isolate variables to solve the problem. Ensure that you are comfortable using your calculator to find tangent values, and remember that solving such systems elegantly combines algebra with trigonometry.

Real-World Problem Solving

Applying right triangle trigonometry to real-world problems like in the exercise demonstrates practical problem-solving skills. This approach is frequently seen in fields such as surveying, construction, and physics where spatial measurements are key.
Real-world problem solving with trigonometry often involves:

• Understanding the practical scenario and translating it into geometric or algebraic forms.
• Accurately drawing the problem and labelling all known values and variables.
• Using mathematical tools, such as trigonometric functions, to solve for unknown values.

In our example, we translated the scenario into equations to determine the tower's height. While solving, consider precision in measurements and calculations, as small errors can significantly impact the results. Practice with diverse scenarios enhances critical thinking and the ability to apply mathematical principles effectively.