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Chapter 1: Problem 68

Height of a Mountain In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is \(3.5^{\circ}\). After you drive 13 miles closer to the mountain, the angle of elevation is \(9^{\circ}\). Approximate the height of the mountain.

Short Answer

Expert verified
Approximate the height of the mountain by solving these equations.

Step by step solution

01

Understand the problem

Here we have two right triangles shared the side which is the height of the mountain. In the first triangle, the angle at the bottom is \(3.5^{\circ}\) and the base is the distance traveled plus the remaining distance to the mountain. In the second triangle, the angle at the bottom is \(9^{\circ}\) and the base is just the remaining distance to the mountain.
02

Apply the formula for tangent and set up the equations

Using the formula for tangent of an angle \( \tan(\theta) = \frac{opposite \ side}{adjacent \ side} \), set up the following two equations: Equation 1: \( Height = \tan(3.5^{\circ}) * (remaining \ distance + 13) \), Equation 2: \( Height = \tan(9^{\circ}) * remaining \ distance \)
03

Solve the system of equations

Subtract equation 2 from equation 1, by doing this we can solve the equation for the remaining distance: \( remaining \ distance = \frac{ \tan(3.5^{\circ}) * 13 }{\tan(9^{\circ}) - \tan(3.5^{\circ})} \), substitute the remaining distance from this equation into either equation 1 or equation 2 give us the height of the mountain.
04

Compute the solution

Now we can plug in the numbers and solve the equations by a calculator to find the exact value of the height of the mountain.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle of Elevation
When trying to measure the height of distant objects like a mountain, one useful tool in trigonometry is the angle of elevation. This angle is measured from the observer鈥檚 line of sight to the horizontal level straight ahead to the topmost point of the object.

Imagine you are standing a certain distance away from a mountain and you want to calculate its height. You would look up to the peak of the mountain, and the angle your line of sight makes with the horizontal is known as the 'angle of elevation'. It's always measured upwards from the horizontal, so if you're standing straight, it's the angle you have to tilt your head back to look at the peak of the mountain.
Tangent of an Angle
The 'tangent of an angle' is a fundamental concept in trigonometry that is used to relate the angles of a right triangle to the lengths of its sides. In a right triangle, the tangent of one of the non-right angles (let's call it \(\theta\)) is the ratio of the length of the side opposite to \(\theta\) to the length of the side adjacent to \(\theta\).

Mathematically, this is expressed as \( \tan(\theta) = \frac{opposite \text{ side}}{adjacent \text{ side}} \). When solving real-world problems involving angles and distances like the height of a mountain, this function is particularly useful because we can rearrange the formula to isolate and calculate the unknown side if we have the angle and the length of one side.
Right Triangle Trigonometry
Right triangle trigonometry is the branch of mathematics that deals with the relationships between the angles and sides of right-angled triangles.

The right triangle has one 90-degree angle, and the trigonometric functions sine, cosine, and tangent are based on the ratios of the sides of right triangles. Specifically, in regards to our mountain height problem, we use the tangent ratio because we are dealing with the angle of elevation and the 'opposite' (height of the mountain) and 'adjacent' (distance from the mountain) sides of the triangle. The right triangle setup allows us to create equations that we can solve for the unknown height.
Solving Trigonometric Equations
Solving trigonometric equations is often required in geometry and trigonometry problems. In our mountain height example, we create two separate equations based on the tangent function for different angles of elevation at different points.

By establishing a system of equations, we can solve for the unknown variables such as the height of the mountain or the distance to it. It's common to manipulate these equations by subtracting one from the other or rearranging terms to isolate one variable. Then, with known values and trigonometric ratios, we can use a calculator to find the numerical values, thus obtaining the height of the mountain.

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Most popular questions from this chapter

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