Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

An airplane flying at an altitude of 10,000 feet passes directly over a fixed object on the ground. One minute later, the angle of depression of the object is \(42^{\circ}\). Approximate the speed of the airplane to the nearest mile per hour.

Step-by-Step Solution

Answer

126 miles per hour.

Step 1: Understanding the problem

We need to calculate the speed of an airplane flying at an altitude of 10,000 feet, given the angle of depression to an object on the ground is \(42^{\circ}\) after one minute. By calculating the ground distance covered in one minute, we can then determine the speed in miles per hour.

Step 2: Relating angles and distances

Since we are dealing with angles and a right triangle (formed by altitude, the angle line, and the ground), we can use trigonometry. The tangent of the angle of depression can be used to relate the altitude to the ground distance. The tangent function is \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Here, the opposite side is 10,000 feet, and the adjacent side is the distance the airplane traveled.

Step 3: Setting up the equation

For the angle of depression \(42^{\circ}\), the equation becomes \( \tan(42^{\circ}) = \frac{10,000}{d} \), where \(d\) is the ground distance covered by the airplane. We need to solve this equation for \(d\).

Step 4: Solving for ground distance

Rearrange the equation to solve for \(d\): \( d = \frac{10,000}{\tan(42^{\circ})} \). Calculating gives approximately \( d \approx \frac{10,000}{0.9004} \approx 11,110.654 \) feet.

Step 5: Converting feet per minute to miles per hour

The calculated distance, 11,110.654 feet, is covered in one minute. To find the speed in miles per hour, convert feet to miles (1 mile = 5280 feet) and then convert the rate from minutes to hours: \( \text{speed} = \frac{11,110.654}{5280} \times 60 \).

Step 6: Calculating final speed

Perform the final calculation: \( \text{speed} = 2.1047 \times 60 \approx 126.282 \). Thus, the airplane's speed is approximately 126 mph to the nearest mile per hour.

Key Concepts

Angles of Depression

In trigonometry, the angle of depression is an essential concept when calculating distances and altitudes. When you look down from a high point to an object on the ground, the angle between your line of sight and the horizontal line is called the angle of depression.

• It is measured from the horizontal line down to the line of sight.
• This concept is useful in solving many real-world problems, such as the one involving the airplane in our exercise.

Understanding angles of depression can help you find the distance between the observer and the object or the object's height if the horizontal distance is known. In our airplane example, the angle of depression helps us relate the altitude of the airplane to the ground distance it covers. Utilizing angles of depression makes it possible to apply trigonometric functions effectively when working with altitude and ground measurements.

Tangent Trigonometric Function

The tangent function is a fundamental part of trigonometry, particularly useful when dealing with right triangles. It relates the ratio of the opposite side to the adjacent side concerning an angle in a right triangle.

• The formula for the tangent function is: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• "Opposite" refers to the side opposite the angle of interest, while "adjacent" refers to the side next to the angle.

For example, in our airplane scenario, the angle of depression allows us to apply the tangent function because a right triangle is formed between the altitude, the distance to the fixed object, and the ground path of the plane.By substituting the known values into the tangent equation, you can solve for the unknown distance traveled along the ground. Thus, tangent functions are crucial for effectively calculating distances or altitudes when working with angles of depression.

Altitude and Distance Calculations

Altitude is the vertical distance an object or point is above a particular reference level, usually the ground. In aviation and many other fields, understanding and being able to calculate altitude and resulting distances is essential.

• In the exercise, the airplane's altitude is given as 10,000 feet, which is the height above the ground.
• Using trigonometric relationships, you can find how far the airplane has traveled over the ground.

Integrating concepts like angles of depression and the tangent function allows for precise altitude and distance calculations. The solution involves setting up and solving an equation where the airplane's altitude is the opposite side of a right triangle, and using the tangent of the angle of depression, we can find the ground distance (adjacent side). This approach is common in solving navigation problems and helps in understanding the geometry involved in flying objects.