Problem 13 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. Sketch an appropriate figure, unless the figure is given. A robot is on the surface of Mars. The angle of depression from a camera in the robot to a rock on the surface of Mars is \(13.33^{\circ} .\) The camera is \(196.0 \mathrm{cm}\) above the surface. How far from the camera is the rock?

Step-by-Step Solution

Answer

The rock is approximately 824.0 cm away from the camera.

Step 1: Understand the Scenario

We have a robot with a camera located 196 cm above the surface of Mars. The camera's angle of depression to a rock on the surface is given as \(13.33^{\circ}\). We need to find the horizontal distance from the camera to the rock.

Step 2: Draw the Scenario

Sketch the scenario, imagining a vertical line from the camera to the surface. The angle of depression is formed between the horizontal line from the camera and the line of sight to the rock. This forms a right triangle with the camera height being one side, the distance to the rock as the adjacent side, and the line of sight as the hypotenuse.

Step 3: Identify the Right Triangle Components

Label the height of the camera as \( AB = 196 \) cm, the horizontal distance to the rock as \( BC \), and the line of sight from the camera to the rock as \( AC \). Given, \( \angle ACB = 13.33^{\circ} \). Here, \( AB \) acts as the opposite side with respect to the angle of depression, and \( BC \) is the adjacent side.

Step 4: Apply Trigonometric Relationship

Use the tangent function which is the ratio of the opposite side to the adjacent side in a right-angled triangle. Therefore, \( \tan(13.33^{\circ}) = \frac{AB}{BC} = \frac{196}{BC} \).

Step 5: Solve the Equation

Rearrange the equation to solve for \( BC \): \[ BC = \frac{196}{\tan(13.33^{\circ})} \] Calculate \( \tan(13.33^{\circ}) \approx 0.2378 \), so, \[ BC = \frac{196}{0.2378} \approx 824.0 \text{ cm} \].

Step 6: Conclusion

The horizontal distance from the camera to the rock, \( BC \), is approximately 824.0 cm.

Key Concepts

Angle of Depression

The angle of depression is a specific angle that occurs when you look down from a point of elevation to a lower object. In problems involving angles of depression, it's vital to understand that this angle is formed between the observer's horizontal line of sight and the line connecting the viewer's eye to the object being viewed. Think of it as the angle at which your line of sight moves downward from the horizontal viewpoint.
The angle of depression is always measured from the horizontal line, not the vertical. In our exercise, the camera on the robot has its lens oriented horizontally, and the angle of depression to the rock on Mars is 13.33 degrees from that horizontal line of sight.
This angle plays a crucial role in setting up our problem as it helps us form the right triangle needed for our calculations.

Tangent Function

The tangent function is an essential part of trigonometry. It provides a relationship between the angles and sides of a right triangle. Specifically, the tangent of an angle (Theta) is the ratio of the length of the opposite side to the length of the adjacent side. Here's the formula:

• \(\tan(\Theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)

In our exercise, we know the angle (13.33^{\circ}) and the opposite side (196 cm, which is the height of the camera). Using the tangent function allows us to find the length of the adjacent side, which in this context is the horizontal distance (BC) to the rock.
This function is particularly useful because it directly relates the known height to the unknown distance, making our calculations straightforward once we insert the given values.

Right Triangle

A right triangle is a triangle in which one angle measures exactly 90 degrees. In problems involving trigonometry, the right triangle is fundamental because it allows us to employ various trigonometric ratios, like sine, cosine, and tangent, depending on the scenario. This specific arrangement aids in solving many geometric problems efficiently.
In the given scenario, the right triangle is formed by the vertical line from the camera to the ground, the horizontal distance from the camera to the rock, and the hypotenuse, which is the line of sight from the camera to the rock.

• The vertical line: Camera height ( AB = 196 cm)
• The horizontal line: Distance to the rock ( BC )
• The hypotenuse: Line of sight ( AC )

This configuration allows us to apply the tangent function, using the known values to find the unknown distance. By visualizing these components, it becomes easier to understand the relationship between the angle of depression, the camera, and the rock on the Martian surface.

Distance Calculation

Distance calculation in trigonometry often involves using a combination of known values and trigonometric identities to find unknown quantities. Here, we're interested in determining the horizontal distance (BC) from the camera to the rock on Mars. Using the tangent function and knowing the camera height and angle of depression simplifies this task.
We start with \(\tan(13.33^{\circ}) = \frac{196}{BC}\). Rearranging gives the formula for BC:

• \(BC = \frac{196}{\tan(13.33^{\circ})}\)

To solve, compute \(\tan(13.33^{\circ})\) (approximately 0.2378) and divide the camera's height by this trigonometric result to find the horizontal distance. The final computation leads us to approximately 824.0 cm.
This method highlights how trigonometry offers practical solutions for calculating distances in right triangle scenarios, especially when direct measurement is challenging or impossible, like in this out-of-this-world problem!