/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 75 Distance A plane flying at an al... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 75

Distance A plane flying at an altitude of 7 miles above a radar antenna will pass directly over the radar antenna (see figure). Let \(d\) be the ground distance from the antenna to the point directly under the plane and let \(x\) be the angle of elevation to the plane from the antenna. ( \(d\) is positive as the plane approaches the antenna.) Write \(d\) as a function of \(x\) and graph the function over the interval \(0

Short Answer

Expert verified
The function that describes \(d\) as a function of \(x\) is \(d = 7 \tan(x)\).

Step by step solution

01

Identify the relationship

First, we need to identify the mathematical relationship between \(d\) and \(x\). From the description, the radar, the plane, and the point directly beneath the plane form a right triangle. We can therefore apply trigonometric rules to establish a relationship between \(d\), \(x\), and the plane's altitude of 7 miles. Because the tangent of an angle in a right triangle is equal to the ratio of the opposite side over the adjacent side, we have \(\tan(x) = \frac{d}{7}\).
02

Isolate d

Then, we鈥檒l need to express \(d\) as a function of \(x\). To do this we solve the above equation for \(d\): multiply both sides by 7 to give \(d = 7 \tan(x)\). This equation expresses \(d\) as a function of \(x\), as requested by the problem.
03

Graph the function

The final task is to graph this function over the interval \(0 < x < \pi\). Since \(\tan(x)\) is a periodic function with period \(\pi\), this will yield a graph depicting the plane's ground distance from the antenna as it approaches and then passes over the antenna, before the distance begins to increase again after the plane has passed.

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Key Concepts

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

Distance Problems
Distance problems are a fascinating application of geometry and trigonometry. These problems often involve finding distances, such as how far an object is from a particular point. In this context, suppose you have a plane flying over a radar antenna at a fixed altitude of 7 miles. The problem asks for the distance from the antenna to the point directly beneath the plane. Understanding distance problems means recognizing the relationship between different components of a triangle involving the object of interest. Here, the plane's altitude serves as one side of a triangle, specifically the opposite side to the angle of elevation. The function - helps to relate the height of the plane to the distance on the ground. - requires recognizing trigonometric relationships鈥攍ike tangent鈥攚hich connect angles and side lengths.
Angle of Elevation
The angle of elevation is a fundamental concept when dealing with objects above a horizontal line of sight. It is measured upwards from the ground or a horizontal plane towards an object, like a plane in the sky. This angle represents the inclination of one's line of sight, and forms a crucial part of our right triangle in this problem. Understanding the angle of elevation: - It can range from 0 to less than 90 degrees. - As the plane gets closer to being directly overhead, this angle increases. Using this angle in distance problems requires translating the inclination into a mathematical term. In our case, the angle is used in trigonometric functions to express distances more clearly.
Tangent Function
The tangent function is one of the essential trigonometric functions. It provides a crucial link between angles and the dimensions of right triangles. Specifically, the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.How it's used in this problem:- Given the triangle in our situation (formed by the radar, the point directly beneath the plane, and the plane), tangent helps calculate the ground distance.- The formula derived here is \[\tan(x) = \frac{d}{7}\]This relationship allows us to express the distance from the radar point directly beneath the plane (\(d\)) using the tangent function as \(d = 7 \tan(x)\). The use of the tangent function is pivotal, as it translates altitude differences into horizontal distances using the angle of elevation.

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Most popular questions from this chapter

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