Problem 58 - Algebra and Trigonometry

Question

Two sensors are spaced 700 feet apart along the approach to a small airport. When an aircraft is nearing the airport, the angle of elevation from the first sensor to the aircraft is \(20^{\circ},\) and from the second sensor to the aircraft it is \(15^{\circ} .\) Determine how high the aircraft is at this time.

Step-by-Step Solution

Answer

116.8 feet

Step 1 - Draw the Situation

Visualize the problem by sketching a triangle. Label the two sensors as points A and B with the distance between them as 700 feet. Let point C be the aircraft. Draw the angles of elevation from both sensors to the aircraft: \( \angle BAC = 20^{\circ} \) and \( \angle ABC = 15^{\circ} \).

Step 2 - Set up the Right Triangles

Notice that we have two right triangles sharing a common vertical side (the height of the aircraft which we need to find). Let's denote the height of the aircraft as h and the horizontal distance from the first sensor (A) to the point directly under the aircraft as x. Thus, the distance from the second sensor (B) to the point directly under the aircraft is (700 - x).

Step 3 - Apply Trigonometry to the First Triangle

Using the right triangle formed by the first sensor: \( \tan(20^{\circ}) = \frac{h}{x} \). Solving for h gives: \( h = x \cdot \tan(20^{\circ}) \).

Step 4 - Apply Trigonometry to the Second Triangle

Using the right triangle formed by the second sensor: \( \tan(15^{\circ}) = \frac{h}{700 - x} \). Solving for h gives: \( h = (700 - x) \cdot \tan(15^{\circ}) \).

Step 5 - Set up the Equation

Since both expressions for h represent the same height, set them equal to each other: \( x \cdot \tan(20^{\circ}) = (700 - x) \cdot \tan(15^{\circ}) \).

Step 6 - Solve for x

Expand and simplify the equation: \( x \cdot \tan(20^{\circ}) = 700 \cdot \tan(15^{\circ}) - x \cdot \tan(15^{\circ}) \). Combine like terms: \( x (\tan(20^{\circ}) + \tan(15^{\circ})) = 700 \cdot \tan(15^{\circ}) \). Solving for x gives: \( x = \frac{700 \cdot \tan(15^{\circ})}{\tan(20^{\circ}) + \tan(15^{\circ})} \). Substitute the values of the tangents: \( x \approx \frac{700 \cdot 0.2679}{0.3640 + 0.2679} \approx 321.33 \text{ feet} \).

Step 7 - Calculate h

Use the value of x in the equation found in Step 3 to find h: \( h = x \cdot \tan(20^{\circ}) \approx 321.33 \cdot 0.3640 \approx 116.8 \text{ feet} \).

Key Concepts

Right triangles

Right triangles play a crucial role in solving many trigonometric problems. A right triangle is a triangle in which one of the angles is exactly 90 degrees. This special property allows us to use trigonometric functions like sine, cosine, and tangent to relate the angles of the triangle to the lengths of its sides. In this problem, we have two right triangles formed by the sensors and the aircraft. Understanding how right triangles work will help you solve problems involving heights and distances. Visualizing the problem by sketching the triangles can be very helpful. Try to draw the hypotenuse (the side opposite the right angle) and identify the angles of elevation and other given dimensions.

Angle of elevation

The angle of elevation is another important concept. It's the angle formed between the horizontal line and the line of sight from an observer to an object above the horizontal. In this exercise, the angles of elevation from two sensors to a point where an aircraft is located are given as 20 degrees and 15 degrees. Knowing how to use this angle is vital for solving the problem, as it connects the observer's position to the aircraft's height. When working with angles of elevation, remember they help create right triangles, where the angles are known and can be used with trigonometric functions.

Trigonometric functions

Trigonometric functions such as sine, cosine, and tangent are fundamental tools in problems involving angles and distances. In this exercise, the primary function used is the tangent (tan). The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side (\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)). Using trigonometric functions correctly involves setting up equations based on the relationships these functions describe. For instance, for the first right triangle with an angle of elevation of 20 degrees, we use \( \tan(20^\text{circ}) = \frac{h}{x} \). Simplifying this helps us find the height (h). Likewise, using the given information from the second sensor, we find another equation and solve. Becoming comfortable with these functions is key to solving trigonometric problems effectively.