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Now the steps to solve the problem: (a) You need to maximize how far the textbook travels horizontally be- fore it reaches an altitude of 0. So first use the formula for h(t) and the quadratic formula to determine at what value of time t the text- book will reach an altitude of 0. Your answer will be in terms of Vvert- (b) Now that you have determined how long the textbook will be in motion (assuming it actually can make it across the river), how far will the textbook travel horizontally before it reaches altitude 0? Your answer will be in terms of Vyert and Vhorz. (c) Use your knowledge of trigonometry to express Vvert and Vhorz in terms of 0. (d) Now express the horizontal disțance the textbook will travel in terms of 0. (e) Solve the optimization problem: which choice of value for 0 will max- imize the horizontal distance the textbook travels? It is okay if your answer has trig or inverse trig functions in it (but it does not need to), but it does need to be a number, in radians (not degrees).

Now the steps to solve the problem: (a) You need to maximize how far the textbook travels horizontally be- fore it reaches an altitude of 0. So first use the formula for h(t) and the quadratic formula to determine at what value of time t the text- book will reach an altitude of 0. Your answer will be in terms of Vvert- (b) Now that you have determined how long the textbook will be in motion (assuming it actually can make it across the river), how far will the textbook travel horizontally before it reaches altitude 0? Your answer will be in terms of Vyert and Vhorz. (c) Use your knowledge of trigonometry to express Vvert and Vhorz in terms of 0. (d) Now express the horizontal disțance the textbook will travel in terms of 0. (e) Solve the optimization problem: which choice of value for 0 will max- imize the horizontal distance the textbook travels? It is okay if your answer has trig or inverse trig functions in it (but it does not need to), but it does need to be a number, in radians (not degrees).

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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to throw the book across the river.2 At what angle should you throw the textbook so that it travels as far as possible before landing, maximizing the odds that it does not land in the water? Here are the relevant details and also some pointers: First, elevation is helping you: your bank of the river is higher than your friend's bank, and there's also your height to consider. The textbook will start out 4 above the elevation of your friends bank, which we'll say has elevation 0. • The textbook will be moving at a total speed of 5 when it leaves your hands. That total speed is related to two other quantities: the rate at which the textbook is rising vertically at time 0, Vvert, and the rate at which it is moving forward horizontally, Vhorz, by the Pythagorean formula 52 = Vert + Vhorz • We will ignore air resistance and assume, per Newton's first law of mo- tion, that the textbook's horizontal speed Vorz is constant through- out its flight. Gravity causes the textbook to accelerate downward at a rate of 10, and solving the initial value problem tells us that the altitude h of the textbook at time t is given by h(t) = 4+Vyert ·t – 5t2
Transcribed Image Text:to throw the book across the river.2 At what angle should you throw the textbook so that it travels as far as possible before landing, maximizing the odds that it does not land in the water? Here are the relevant details and also some pointers: First, elevation is helping you: your bank of the river is higher than your friend's bank, and there's also your height to consider. The textbook will start out 4 above the elevation of your friends bank, which we'll say has elevation 0. • The textbook will be moving at a total speed of 5 when it leaves your hands. That total speed is related to two other quantities: the rate at which the textbook is rising vertically at time 0, Vvert, and the rate at which it is moving forward horizontally, Vhorz, by the Pythagorean formula 52 = Vert + Vhorz • We will ignore air resistance and assume, per Newton's first law of mo- tion, that the textbook's horizontal speed Vorz is constant through- out its flight. Gravity causes the textbook to accelerate downward at a rate of 10, and solving the initial value problem tells us that the altitude h of the textbook at time t is given by h(t) = 4+Vyert ·t – 5t2
Now the steps to solve the problem: (a) You need to maximize how far the textbook travels horizontally be- fore it reaches an altitude of 0. So first use the formula for h(t) and the quadratic formula to determine at what value of time t the text- book will reach an altitude of 0. Your answer will be in terms of Vvert · (b) Now that you have determined how long the textbook will be in motion (assuming it actually can make it across the river), how far will the textbook travel horizontally before it reaches altitude 0? Your answer will be in terms of Vyert and Vnorz. (c) Use your knowledge of trigonometry to express Vvert and Vhorz in terms of 0. (d) Now express the horizontal disțance the textbook will travel in terms of 0. (e) Solve the optimization problem: which choice of value for 0 will max- imize the horizontal distance the textbook travels? It is okay if your answer has trig or inverse trig functions in it (but it does not need to), but it does need to be a number, in radians (not degrees). 2Why would someone fling a textbook across a river? Don't ask me; I'm not the one hurling books!
Transcribed Image Text:Now the steps to solve the problem: (a) You need to maximize how far the textbook travels horizontally be- fore it reaches an altitude of 0. So first use the formula for h(t) and the quadratic formula to determine at what value of time t the text- book will reach an altitude of 0. Your answer will be in terms of Vvert · (b) Now that you have determined how long the textbook will be in motion (assuming it actually can make it across the river), how far will the textbook travel horizontally before it reaches altitude 0? Your answer will be in terms of Vyert and Vnorz. (c) Use your knowledge of trigonometry to express Vvert and Vhorz in terms of 0. (d) Now express the horizontal disțance the textbook will travel in terms of 0. (e) Solve the optimization problem: which choice of value for 0 will max- imize the horizontal distance the textbook travels? It is okay if your answer has trig or inverse trig functions in it (but it does not need to), but it does need to be a number, in radians (not degrees). 2Why would someone fling a textbook across a river? Don't ask me; I'm not the one hurling books!
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