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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.R.115
Chapter 3, Problem 3.R.115

A jet flying at 450 mi/hr and traveling in a straight line at a constant elevation of 500 ft passes directly over a spectator at an air show. How quickly is the angle of elevation (between the ground and the line from the spectator to the jet) changing 2 seconds later? 

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First, understand the scenario: The jet is flying horizontally at a constant speed of 450 mi/hr, which needs to be converted to feet per second for consistency with the elevation given in feet. Use the conversion factor: 1 mile = 5280 feet.
Next, set up the relationship between the angle of elevation \( \theta \), the horizontal distance \( x \) from the spectator to the jet, and the constant elevation \( h = 500 \) ft. The tangent of the angle \( \theta \) is given by \( \tan(\theta) = \frac{h}{x} \).
Differentiate the equation \( \tan(\theta) = \frac{h}{x} \) with respect to time \( t \) to find \( \frac{d\theta}{dt} \). Use implicit differentiation: \( \sec^2(\theta) \cdot \frac{d\theta}{dt} = -\frac{h}{x^2} \cdot \frac{dx}{dt} \).
Determine \( \frac{dx}{dt} \), the rate at which the horizontal distance \( x \) is changing. Since the jet is moving at 450 mi/hr, convert this speed to feet per second: \( \frac{450 \times 5280}{3600} \) ft/s.
Finally, substitute the known values into the differentiated equation. At \( t = 2 \) seconds, calculate \( x \) using \( x = \text{speed} \times t \). Then, solve for \( \frac{d\theta}{dt} \) using the values of \( x \), \( h \), and \( \frac{dx}{dt} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to determine how the angle of elevation changes as the jet moves horizontally. This requires applying the concept of derivatives to relate the rates of change of the angle and the horizontal distance traveled by the jet.
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Trigonometric Functions

Trigonometric functions, particularly tangent in this context, relate angles to the ratios of sides in right triangles. The angle of elevation can be expressed using the tangent function, where the opposite side is the height of the jet and the adjacent side is the horizontal distance from the spectator to the jet. Understanding how to manipulate these functions is crucial for solving the problem.
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Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of a quantity. In this scenario, we will differentiate the expression for the angle of elevation with respect to time to find how quickly the angle is changing. This requires applying implicit differentiation to relate the changes in the angle to the changes in the horizontal distance and height.
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