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How fast is the distance between the range finder and the balloon increasing (in meters per minute), when the range finder's elevation angle is 4 radians ? Enter a numerical answer, rounded to two decimal places.

How fast is the distance between the range finder and the balloon increasing (in meters per minute), when the range finder's elevation angle is 4 radians ? Enter a numerical answer, rounded to two decimal places.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Carefully read Example 2 on pages 192-193, and answer the following related question: How fast is the distance between the range fınder and the balloon increasing (in meters per minute), when the range finder's elevation angle is radians ? Enter a numerical answer, rounded to two decimal places.
Transcribed Image Text:Carefully read Example 2 on pages 192-193, and answer the following related question: How fast is the distance between the range fınder and the balloon increasing (in meters per minute), when the range finder's elevation angle is radians ? Enter a numerical answer, rounded to two decimal places.
A hot air balloon rising straight up from a level field is tracked by a range finder 150 m from the liftoff point. At the moment the range finder's elevation angle is T/4, the angle is increasing at the rate of 0.14 rad /min. How fast is the balloon rising at EXAMPLE 2 that moment? 3.10 Related Rates 193 Solution We answer the question in the six strategy steps. Balloon 1. Draw a picture and name the variables and constants (Figure 3.46). The variables in the picture are de = 0.14 rad/min 0 = the angle in radians the range finder makes with the ground. dt dy = ? dt when e = T/4 y = the height in meters of the balloon above the ground. when e = 7/4 We let t represent time in minutes and assume that 0 and y are differentiable functions of t. The one constant in the picture is the distance from the range finder to the liftoff point (150 m). There is no need to give it a special symbol. Range finder 150 m 2. Write down the additional numerical information. FIGURE 3.46 The rate of change of the balloon's height is related to the rate of do TT = 0.14 rad/min dt when 4 change of the angle the range finder makes with the ground (Example 2). 3. Write down what we are to find. We want dy/dt when 0 = T/4. 4. Write an equation that relates the variables y and 0. y = tan 0 y = 150 tan 0 or 150 5. Differentiate with respect to t using the Chain Rule. The result tells how dy/dt (which we want) is related to de /dt (which we know). dy = 150 (sec? 0) dt do dt 6. Evaluate with 0 = /4 and d0/dt = 0.14 to find dy/dt. dy 150( V2) (0.14) = 42 sec dt At the moment in question, the balloon is rising at the rate of 42 m/min.
Transcribed Image Text:A hot air balloon rising straight up from a level field is tracked by a range finder 150 m from the liftoff point. At the moment the range finder's elevation angle is T/4, the angle is increasing at the rate of 0.14 rad /min. How fast is the balloon rising at EXAMPLE 2 that moment? 3.10 Related Rates 193 Solution We answer the question in the six strategy steps. Balloon 1. Draw a picture and name the variables and constants (Figure 3.46). The variables in the picture are de = 0.14 rad/min 0 = the angle in radians the range finder makes with the ground. dt dy = ? dt when e = T/4 y = the height in meters of the balloon above the ground. when e = 7/4 We let t represent time in minutes and assume that 0 and y are differentiable functions of t. The one constant in the picture is the distance from the range finder to the liftoff point (150 m). There is no need to give it a special symbol. Range finder 150 m 2. Write down the additional numerical information. FIGURE 3.46 The rate of change of the balloon's height is related to the rate of do TT = 0.14 rad/min dt when 4 change of the angle the range finder makes with the ground (Example 2). 3. Write down what we are to find. We want dy/dt when 0 = T/4. 4. Write an equation that relates the variables y and 0. y = tan 0 y = 150 tan 0 or 150 5. Differentiate with respect to t using the Chain Rule. The result tells how dy/dt (which we want) is related to de /dt (which we know). dy = 150 (sec? 0) dt do dt 6. Evaluate with 0 = /4 and d0/dt = 0.14 to find dy/dt. dy 150( V2) (0.14) = 42 sec dt At the moment in question, the balloon is rising at the rate of 42 m/min.
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