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Chapter 6: Problem 56

An airplane is flying at an elevation of 5150 ft, directly above a straight highway. Two motorists are driving cars on the highway on opposite sides of the plane, and the angle of depression to one car is \(35^{\circ}\) and to the other is \(52^{\circ} .\) How far apart are the cars?

Short Answer

Expert verified
The cars are approximately 11379.5 ft apart.

Step by step solution

01

Understanding the Problem

We need to find the distance between two cars driving on a highway using angles of depression from an airplane flying directly above. The angles of depression given are \(35^{\circ}\) and \(52^{\circ}\).
02

Define the Geometry

Visualize the problem as a right triangle, where the airplane is at the vertical position with a height (altitude) of 5150 ft above the highway. The angles of depression \(35^{\circ}\) and \(52^{\circ}\) form right triangles with the cars on the ground.
03

Set Up Right Triangles

For each car, we can form a right triangle. The angle of depression equals the angle of elevation from the car due to alternative interior angles.
04

Use Trigonometric Functions

For each triangle, use the tangent function. For the angle \(35^{\circ}\), the equation is: \(\tan(35^{\circ}) = \frac{5150}{d_1}\), where \(d_1\) is the distance from plane to the first car. For the angle \(52^{\circ}\), the equation is: \(\tan(52^{\circ}) = \frac{5150}{d_2}\), where \(d_2\) is the distance from plane to the second car.
05

Solve for Distances

Solve for \(d_1\) and \(d_2\) using the tangent equations: \(d_1 = \frac{5150}{\tan(35^{\circ})}\) and \(d_2 = \frac{5150}{\tan(52^{\circ})}\). Calculate the values using a calculator.
06

Calculate Values

Compute \(d_1\) and \(d_2\). Using a calculator, \( \tan(35^{\circ}) \approx 0.7002\) thus \(d_1 \approx \frac{5150}{0.7002} \approx 7355.4 \text{ ft}\). For \( \tan(52^{\circ}) \approx 1.2799\), \(d_2 \approx \frac{5150}{1.2799} \approx 4024.1 \text{ ft} \).
07

Find the Distance Between Cars

The total distance between the cars is the sum of \(d_1\) and \(d_2\). Therefore, the distance is approximately \(7355.4 + 4024.1 = 11379.5 \text{ ft} \).

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Key Concepts

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

Understanding Angles of Depression
When learning about angles of depression, imagine you are seated high up, looking down at an object on the ground. The angle of depression is the angle formed between your line of sight and the horizontal line drawn level to your eye. It is always the same as the angle of elevation from the object on the ground looking up toward you.
- This principle is useful when calculating distances, as it maintains a correspondence between angles up and down.
- In our exercise, the airplane鈥檚 perspective results in angles of depression at both 35掳 and 52掳. The concept helps you set up geometric relationships required to solve trigonometric problems in the real world. By knowing the elevation and angle, we can apply trigonometric formulas to find distances, like how far apart two cars are on the ground.
Right Triangles in Trigonometry
A right triangle is the building block of many trigonometric calculations. In a right triangle, one angle is always 90掳, and the other two are less than 90掳. Understanding right triangles is crucial for solving problems involving angles and distances.
- In our problem, each angle of depression creates a right triangle, with the height of 5150 ft as one side of the triangle. - Recognizing these right triangles lets us employ trigonometric ratios like tangent to find unknown distances. The right triangles formed from each angle of depression allow us to harness these relationships. This logic immensely simplifies real-world problems by converting them into manageable geometric shapes, each step then follows almost naturally from the previous.
Utilizing the Tangent Function
The tangent function is key to solving problems involving angles and right triangles. It relates the angle in a triangle to the lengths of two sides. Specifically:
- \( an(\theta) = \frac{\text{opposite}}{\text{adjacent}}\) - In our context, for angle of depression and right triangles, 'opposite' refers to the height of the airplane (5150 ft), while 'adjacent' is the distance from the airplane's point to the car on the highway.Using the tangent function for angles of 35掳 and 52掳, we solve for the distances along the highway to each car.With the formula set as \( an(35掳) = \frac{5150}{d_1}\) and \( an(52掳) = \frac{5150}{d_2}\), the tangent function's inversions provide easy access to otherwise challenging measurements.
This applies across various fields where triangulation is used for measurement, from navigation to architecture.

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