Problem 59 - Algebra and Trigonometry

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 estimated height of the mountain is approximately 1000 feet.

Step 1: Setup the problem

This is a trigonometry problem where we want to calculate the height of the mountain using the angles of elevation and the known distance between the two points where the angles were measured. Let's denote the initial distance from the mountain as \(x\) feet, and the height of the mountain as \(h\).

Step 2: Use the tangent function

For both points along the plain, we can use the tangent of the angle of elevation. At the first point, we have:\[\tan(32^{\circ}) = \frac{h}{x}\]And at the closer point, 1000 feet nearer to the mountain:\[\tan(35^{\circ}) = \frac{h}{x - 1000}\]

Step 3: Solve the first equation for h

In the first equation, solve for \(h\):\[h = x \cdot \tan(32^{\circ})\]

Step 4: Solve the second equation for h

In the second equation, solve for \(h\):\[h = (x - 1000) \cdot \tan(35^{\circ})\]

Step 5: Equate the expressions for h

Since both expressions equal \(h\), equate them:\[x \cdot \tan(32^{\circ}) = (x - 1000) \cdot \tan(35^{\circ})\]

Step 6: Solve for x

Rearrange the equation to solve for \(x\):\[x \cdot \tan(32^{\circ}) = x \cdot \tan(35^{\circ}) - 1000 \cdot \tan(35^{\circ})\]\[x (\tan(32^{\circ}) - \tan(35^{\circ})) = -1000 \cdot \tan(35^{\circ})\]\[x = \frac{-1000 \cdot \tan(35^{\circ})}{\tan(32^{\circ}) - \tan(35^{\circ})}\]

Step 7: Calculate h using x

Once we have \(x\), substitute it back into one of the equations for \(h\), for example:\[h = x \cdot \tan(32^{\circ})\]Calculate this value to find the height of the mountain.

Key Concepts

Angle of Elevation

The angle of elevation is a fundamental concept in trigonometry. It refers to the angle between the horizontal line from the observer's eye to an object above the observer. In simpler terms, imagine looking straight ahead, then tilting your head upwards to see the top of a mountain. The angle your head tilts upward is the angle of elevation.

In our mountain problem, measurements are taken from two different points on the plain. Each measurement gives us a different angle of elevation—the first at \(32^{\circ}\) and the second, closer to the mountain, at \(35^{\circ}\).

• An angle of elevation tells us how steeply we must look up from a point on the ground to see the top of the mountain.
• Elevating from different positions can give different angles based on proximity to the object.

Knowing these angles helps us use trigonometric relationships to estimate distances or heights, which are not easily measured directly.

Tangent Function

The tangent function is a key tool in trigonometry for solving problems related to angles and lengths. It's part of the primary trigonometric functions and is defined as the ratio of the opposite side to the adjacent side in a right triangle. In mathematical terms, \( an(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).

In the mountain scenario, the tangent function is used with the angles of elevation to express the height \(h\) of the mountain in relation to the horizontal distance \(x\) from the observation point to the base of the mountain.

• At the first location: \[ \tan(32^{\circ}) = \frac{h}{x} \]
• At the second, closer location: \[ \tan(35^{\circ}) = \frac{h}{x - 1000} \]

These equations represent how the same height relates differently to each distance due to the change in angle. By utilizing them, we can solve for unknown distances or heights when direct measurement is impractical.

Problem Solving

Problem-solving in trigonometry often involves setting up a scenario using known quantities and relationships to find unknown variables. In this exercise, our goal is to find the mountain's height using angles of elevation and trigonometric functions.

The process typically involves:

• Understanding what information you have and what is needed.
• Setting up equations based on trigonometric functions like tangent.
• Solving step by step, checking each calculation.

Here, after establishing equations for the height from each observation point, we equated them because they both represent \(h\). Solving the simultaneous equations lets us find \(x\), and then \(h\). This structured approach is common in trigonometry: define the problem, express it using known functions, and solve for the unknown.
With practice, these steps become intuitive, helping you tackle a variety of real-world trigonometric questions.