Problem 58 - Precalculus Mathematics for Calculus

Question

A hot-air balloon is floating above a straight road. To estimate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be \(20^{\circ}\) and \(22^{\circ}\). How high is the balloon?

Step-by-Step Solution

Answer

The balloon is approximately 3.68 miles above the ground.

Step 1: Understanding Angles of Depression

The angle of depression is the angle between the horizontal line from the balloon to the line of sight towards a point on the ground. Hence, if you extend a horizontal line from the balloon, this angle is between the horizontal line and the line of sight down to the milepost.

Step 2: Establishing the Right Triangles

Visualize two right triangles formed by the line of sight from the balloon to each of the mileposts. Let's denote the height as \( h \), the distance to the milepost with the \( 22^{\circ} \) angle of depression as \( x \), and to the \( 20^{\circ} \) angle of depression as \( x + 1 \).

Step 3: Using the Tangent Function for First Milepost

The tangent of an angle in a right triangle is the opposite side over the adjacent side. For the \( 22^{\circ} \) depression angle, \( \tan(22^{\circ}) = \frac{h}{x} \). So, we have \( h = x \tan(22^{\circ}) \).

Step 4: Using the Tangent Function for Second Milepost

Similarly, for the \( 20^{\circ} \) angle of depression, \( \tan(20^{\circ}) = \frac{h}{x+1} \). So, \( h = (x+1) \tan(20^{\circ}) \).

Step 5: Equating the Two Expressions for Height

From both equations for \( h \), we have: \[ x \tan(22^{\circ}) = (x + 1) \tan(20^{\circ}) \].This equation allows us to solve for \( x \).

Step 6: Solving for x

Rearrange the equation: \[ x \tan(22^{\circ}) = x \tan(20^{\circ}) + \tan(20^{\circ}) \].Simplify and solve for \( x \):\[ x (\tan(22^{\circ}) - \tan(20^{\circ})) = \tan(20^{\circ}) \].Thus: \[ x = \frac{\tan(20^{\circ})}{\tan(22^{\circ}) - \tan(20^{\circ})} \].

Step 7: Calculating x and Evaluating h

Calculate \( x \) using trigonometric values, then substitute back to find \( h \):Using a calculator: \( \tan(20^{\circ}) \approx 0.364 \) and \( \tan(22^{\circ}) \approx 0.404 \).Thus, \[ x \approx \frac{0.364}{0.404 - 0.364} \approx 9.1 \] miles.Then, calculate \( h \) using one of the expressions, say \( h = x \tan(22^{\circ}) \): \( h \approx 9.1 \cdot 0.404 \approx 3.68 \) miles.

Key Concepts

Angles of Depression

When you hear "angle of depression," it might sound like something that only mathematicians love talking about, but it's quite easy to understand! Imagine you are in a hot-air balloon floating in the sky. You look straight ahead horizontally; this is known as your line of the horizon. Now, if you look down towards the ground, say at a milepost directly below, you form what is called an angle of depression. This angle is formed between your horizontal line of sight and the line down to the object.

The angle of depression is crucial in solving height-related problems because it provides the angular measurement needed to form a right triangle. Knowing whether the problem involves angles of depression can help you visualize the scenario and set up equations effectively, especially when using trigonometric functions. So, next time you're gazing down from above, just remember: that angle can help you find how high you really are!

Right Triangles

Right triangles are like the fundamental building blocks in trigonometry. They consist of one right angle (that’s 90 degrees) and two acute angles, which together always add up to 180 degrees.

In trigonometry problems, such as the balloon scenario, right triangles help you connect angular and linear measurements. When dealing with angles of depression, visualizing how a right triangle forms between the observer’s line of sight and the ground is important. In our example, the height of the balloon is the opposite side of the triangle, while the distance to the milepost represents the adjacent side.

Understanding this setup allows you to apply trigonometric functions like tangent effectively, as they rely on these side-to-angle relationships. With right triangles, you can decode complex situations into simpler, solvable equations.

Tangent Function

The tangent function is one of the core tools in understanding and solving trigonometry problems involving right triangles. In any right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

For an angle \(\theta\), the formula is given by:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

In the context of the balloon problem, the angles of depression of \(20^{\circ}\) and \(22^{\circ}\) allow us to use the tangent function to create equations involving the balloon's height \(h\) and the distances \(x\) and \(x+1\). By knowing \(\tan(22^{\circ})\) and \(\tan(20^{\circ})\), you can set up two separate expressions that both equal \(h\), the balloon's height.

These equations can then be solved to find \(h\), showing how the tangent function plays a pivotal role in linking angles to lengths in right triangle trigonometry.