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Math
CalculusSingle Variable Calculus: Early Transcendentals, Volume I8th Edition

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Chapter 4.7, Problem 75E
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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

An observer stands at a point P, one unit away from a track. Two runners start at the point S in the figure and ran along the track. One runner runs three times as fast as the other. Find the maximum value of the observer’s angle of sight θ between the runners.

Chapter 4.7, Problem 75E, An observer stands at a point P, one unit away from a track. Two runners start at the point S in the

To determine

To find: The maximum value of the observer’s angle of sight θ between the runners.

Explanation

Given:

Distance between the point P from the track =1

Calculation:

Two runners start at the point S and run along the track.

One runner runs three times as fast as the other.

Let the distance of BC=3t.

Consider the triangle PSD,

Let, ∠P=ψ

Consider the triangle PCD

∠P=θ

Consider the triangle PSC

∠P=θ+ψ

Then,

tan(θ+ψ)=3t1tanθ+tanψ1−tanθtanψ=3tt+tanθ1−ttanθ=3t

Simplify further,

3t(1−ttanθ)=t+tanθ2t=(1+3t2)tanθtanθ=2t1+3t2

Let f(t)=tanθ=2t1+3t2

Differentiate with respect to t,

dfdt=2(1+3t2)−2t(6t)(1+3t2)2=2(1−3t2)(1+3t2)2

For critical points, dfdt=0

2(1−3t2)(1+3t2)2=01−3t2=0t=±13

Differentiate dfdt with respect to t,

d2fdt2=(1+3t2)2(−12t)−2(1−

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63. Revenue Revenue R from the sale of x units of a product is found by multiplying the price by the number o...

Mathematical Applications for the Management, Life, and Social Sciences

There is considerable evidence to support the theory that for some species there is a minimum population m such...

Single Variable Calculus: Early Transcendentals

Perform the indicted operation and simplify: 316(214)

Elementary Technical Mathematics

Prove that the equalities in Exercises hold for all in . Assume only the basic postulates for and those prop...

Elements Of Modern Algebra

In Exercises 27 to 32, use SSS, SAS, ASA, or AAS to prove that the triangles are congruent. Given: PN bisects M...

Elementary Geometry For College Students, 7e

Give the reciprocal of each number. x

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Set up and solve equations for the following business situations. At Trident Sporting Goods, the inventory rati...

Contemporary Mathematics for Business & Consumers

Finding a Maclaurin Series In Exercises 27-40, find the Maclaurin series for the function. Use the table of pow...

Calculus of a Single Variable

Proof Prove that if limxcf(x) exists and limxc[f(x)+g(x)] does not exist, then limxcg(x) does not exist.

Calculus: Early Transcendental Functions (MindTap Course List)

Find the accumulated amount after 8 years if 12,000 is invested at 3.5/year compounded a annually, b semi-annua...

Finite Mathematics for the Managerial, Life, and Social Sciences

In Exercises 27 to 32, use a calculator and reciprocal relationships to find each ratio correct to four decimal...

Elementary Geometry for College Students

Precalculus or Calculus In Exercises 3-6.decide whether the problem can be solved using precalculus or whether ...

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True or False: The Mean Value Theorem for Integrals is a restatement of the Mean Value Theorem (of Chapter 4) b...

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Finding the domain In exercises 310 find the domain of the vector valued function. r(t)=F(t)G(t),where. F(t)=t3...

Multivariable Calculus

Using a binomial series, the Maclaurin series for is:

Study Guide for Stewart's Multivariable Calculus, 8th

African Elephant Populations Although millions of elephants once roamed across Africa, by the mid-1980s elephan...

Statistics for Business & Economics, Revised (MindTap Course List)

Using PsycINFO (or a similar database), find research articles on how background music influences mood. Print o...

Research Methods for the Behavioral Sciences (MindTap Course List)

Distinguish between a hypothesis and a prediction.

Research Methods for the Behavioral Sciences (MindTap Course List)

Express each value as a percent. 27. 159100

Mathematics For Machine Technology

Suppose N = 15 and r = 4. What is the probability of x = 3 for n = 10?

STATISTICS F/BUSINESS+ECONOMICS-TEXT

In Exercises 95-98, match the equation of the parabola with its graph. y2=9x a. b. c. d.

College Algebra (MindTap Course List)

The accompanying table gives the mean and standard deviation of reaction times (in seconds) for each of two dif...

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Modeling Data The table shows the gross income tax collections (in billions of dollars) by the Internal Revenue...

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