Problem 59 - Precalculus Mathematics for Calculus

Question

To estimate the height of a mountain above a level plain, the angle of elevation to the top of the mountain is measured to be \(32^{\circ} .\) One thousand feet closer to the mountain along the plain, it is found that the angle of elevation is \(35^{\circ} .\) Estimate the height of the mountain.

Step-by-Step Solution

Answer

The mountain is approximately 1797 feet high.

Step 1: Understanding the Problem

We need to find the height of a mountain by using trigonometric methods given two angles of elevation and a horizontal distance between them.

Step 2: Drawing the Diagram

Visualize the scenario with two right triangles. Place two observation points on the horizontal ground with known distance apart, and measure angles of elevation from these points to the top of the mountain.

Step 3: Defining Variables and Known Values

Let \(h\) be the height of the mountain. Let \(d_1\) be the distance from the first point to the mountain and \(d_2 = d_1 + 1000\) ft be the distance from the second point. The angles given are \(32^{\circ}\) and \(35^{\circ}\).

Step 4: Using Trigonometric Functions

For the first triangle, \(\tan(32^{\circ}) = \frac{h}{d_2}\). For the second triangle, \(\tan(35^{\circ}) = \frac{h}{d_1}\). These equations represent the tangent of the angles in terms of \(h\) and the distances.

Step 5: Setting Up Equations

From \(\tan(32^{\circ}) = \frac{h}{d_2}\), we have \(h = d_2 \cdot \tan(32^{\circ})\). From \(\tan(35^{\circ}) = \frac{h}{d_1}\), we have \(h = d_1 \cdot \tan(35^{\circ})\).

Step 6: Equating and Solving for Distances

Set \(d_2 \cdot \tan(32^{\circ}) = d_1 \cdot \tan(35^{\circ})\). Since \(d_2 = d_1 + 1000\), substitute and solve for \(d_1\).

Step 7: Calculate \(d_1\)

\((d_1 + 1000) \cdot \tan(32^{\circ}) = d_1 \cdot \tan(35^{\circ})\). This simplifies to calculate \(d_1\).

Step 8: Determine the mountain height

Use the determined \(d_1\) to find \(h = d_1 \cdot \tan(35^{\circ})\) or \(h = (d_1 + 1000) \cdot \tan(32^{\circ})\).

Key Concepts

Angle of Elevation

The **Angle of Elevation** is a useful concept when estimating the height of an object, like a mountain, from a distance. Imagine standing at a certain point and looking up towards the peak of a mountain. The angle formed between the line of sight—drawn straight from the eye to the mountain top—and the horizontal ground line is called the angle of elevation.

In our problem, two angles of elevation are considered: one observed from a point 1000 feet closer to the mountain and another from a more distant point. These angles measure at 32° and 35°, respectively. A larger angle of elevation usually indicates closer proximity in a horizontal direction. By using these angles, we can determine unknown heights using trigonometric functions effectively.

Tangent Function

Trigonometry inherently involves relationships between angles and sides of triangles, especially right triangles. The **Tangent Function** is one such relationship. In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

If we denote the height of the mountain as \(h\) and the horizontal distances from the points of observations as \(d_1\) and \(d_2\), then:

• \(\tan(32^{\circ}) = \frac{h}{d_2}\)
• \(\tan(35^{\circ}) = \frac{h}{d_1}\)

These equations help us express the relationship between the angle of elevation and the known distances. The tangent values of known angles help estimate the unknown height by rearranging the equations.

Right Triangle

A **Right Triangle** is a triangle with one of its angles measuring exactly 90°. In trigonometric problems like the one discussed, right triangles provide the basis for establishing relationships between various trigonometric functions and triangle sides.

The scenario of estimating a mountain's height by using angles of elevation is visualized through right triangles. The mountain is perpendicular to the base, making the setup a classic example of a right triangle problem. This helps because trigonometric functions like sine, cosine, and tangent are all based on the properties of right triangles.

The two observation points and the mountain form two separate right triangles with different angles of elevation. Thus, the height estimation heavily relies on understanding these triangles and using trigonometric identities effectively.

Height Estimation

**Height Estimation** involves calculating the actual height of an object by using indirect measurements and mathematical concepts. In this exercise, it was necessary to estimate the height of a mountain by applying trigonometry to two right triangles formed by two observation points on the plain.

Here’s how height estimation unfolds:

• Use given angles of elevation to establish equations: \(\tan(32^{\circ})\) and \(\tan(35^{\circ})\) relate height \(h\) to distances \(d_1\) and \(d_2.\)
• Rearrange these to solve for one unknown variable: \(h = d_1 \cdot \tan(35^{\circ})\) or \(h = (d_1 + 1000) \cdot \tan(32^{\circ})\).
• Use the equations to find distances and height by substitution and rearranging methods.

Height estimation through trigonometry allows solving complex real-world measurements where direct measurement is not feasible.