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Math
CalculusSingle Variable Calculus: Early Transcendentals, Volume I8th Edition

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Chapter 4.7, Problem 82E
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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
Problem 10E
Problem 11E
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Problem 13E
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Problem 15E
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Problem 18E
Problem 19E
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Problem 21E
Problem 22E
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Problem 44E
Problem 45E
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Problem 55E
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Problem 69E
Problem 70E
Problem 71E
Problem 72E
Problem 73E
Problem 74E
Problem 75E
Problem 76E
Problem 77E
Problem 78E
Problem 79E
Problem 80E
Problem 81E
Problem 82E
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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line ℓ, parallel to the line joining the light sources and at a distance d meters from it (see the figure). We want to locate P on ℓ. so that the intensity of illumination is minimized. We need to use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source.

(a) Find an expression for the intensity I(x) at the point P.

(b) If d = 5 m, use graphs of I(x) and I′(x) to show that the intensity is minimized when x = 5 m, that is, when P is at the midpoint of ℓ.

(c) If d = 10 m, show that the intensity (perhaps surprisingly) is not minimized at the midpoint.

(d) Somewhere between d = 5 m and d = 10 m there is a transitional value of d at which the point of minimal illumination abruptly changes. Estimate this value of d by graphical methods. Then find the exact value of d.

Chapter 4.7, Problem 82E, Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point

(a)

To determine

To find: The expression for the intensity I(x) at the point P.

Explanation

Given:

Distance between two light source = 10m.

Distance between the point P and the line joining light source = d.

Formula used:

Pythagorean Theorem:

If c denotes the length of the hypotenuse and a and b denote the length of the other two sides of a right angle triangle, then the Pythagorean theorem can be expressed as the Pythagorean equation: a2+b2=c2

Calculation:

Let the strength of the source be k.

Let the intensity on the point P be I.

In Figure 1,

The light sources are at the points A and B.

I(x)∝strength of source(distance from source)2.

By Pythagorean Theorem on the triangle ΔPAC,

PA2=x2+d2.

Hence the intensity due to the light source at the points A is kx2+d2

By Pythagorean Theorem on the triangle ΔPBC,

PB2=(10−x)2+d2

(b)

To determine

To show: The intensity is minimized when x=5, that is when P is at the midpoint l.

(c)

To determine

To show: The intensity is not minimized when d=10 at the midpoint l.

(d)

To determine

To estimate: The value of d.

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Sect-4.1 P-1ESect-4.1 P-2ESect-4.1 P-3ESect-4.1 P-4ESect-4.1 P-5ESect-4.1 P-6ESect-4.1 P-7ESect-4.1 P-8ESect-4.1 P-9ESect-4.1 P-10E
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P-1ESect-4.2 P-2ESect-4.2 P-3ESect-4.2 P-4ESect-4.2 P-5ESect-4.2 P-6ESect-4.2 P-7ESect-4.2 P-8ESect-4.2 P-9ESect-4.2 P-10ESect-4.2 P-11ESect-4.2 P-12ESect-4.2 P-13ESect-4.2 P-14ESect-4.2 P-15ESect-4.2 P-16ESect-4.2 P-17ESect-4.2 P-18ESect-4.2 P-19ESect-4.2 P-20ESect-4.2 P-21ESect-4.2 P-22ESect-4.2 P-23ESect-4.2 P-24ESect-4.2 P-25ESect-4.2 P-26ESect-4.2 P-27ESect-4.2 P-28ESect-4.2 P-29ESect-4.2 P-30ESect-4.2 P-31ESect-4.2 P-32ESect-4.2 P-33ESect-4.2 P-34ESect-4.2 P-36ESect-4.2 P-37ESect-4.2 P-38ESect-4.3 P-1ESect-4.3 P-2ESect-4.3 P-3ESect-4.3 P-4ESect-4.3 P-5ESect-4.3 P-6ESect-4.3 P-7ESect-4.3 P-8ESect-4.3 P-9ESect-4.3 P-10ESect-4.3 P-11ESect-4.3 P-12ESect-4.3 P-13ESect-4.3 P-14ESect-4.3 P-15ESect-4.3 P-16ESect-4.3 P-17ESect-4.3 P-18ESect-4.3 P-19ESect-4.3 P-20ESect-4.3 P-21ESect-4.3 P-22ESect-4.3 P-23ESect-4.3 P-24ESect-4.3 P-25ESect-4.3 P-26ESect-4.3 P-27ESect-4.3 P-28ESect-4.3 P-29ESect-4.3 P-30ESect-4.3 P-31ESect-4.3 P-32ESect-4.3 P-33ESect-4.3 P-34ESect-4.3 P-35ESect-4.3 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Schmidt (1994) conducted a series of experiments examining the effects of humor on memory. In one study, partic...

Statistics for The Behavioral Sciences (MindTap Course List)

If r is a rational function, use Exercise 57 to show that limxa r(x) = r(a) for every number a in the domain of...

Single Variable Calculus

The article cited in Example 1.2 also gave the accompanying strength observations for cylinders: a. Construct a...

Probability and Statistics for Engineering and the Sciences

In problems 23-58, perform the indicated operations and simplify. 48.

Mathematical Applications for the Management, Life, and Social Sciences

In Exercises 71 to 82, determine whether each of the sets is a well-defined set. The set of small countries

Mathematical Excursions (MindTap Course List)

In Exercises 25 and 26, find the line that best fits the given points, and compute the correlation coefficient....

Finite Mathematics

For Problems 55-64, answer the question with an algebraic expression. Yuriko can type w words in an hour. What ...

Intermediate Algebra

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves abou...

Single Variable Calculus: Early Transcendentals

Follow the rules for working with measurements. Find the volume of each figure in Exercises 110:

Elementary Technical Mathematics

True or false Label each of the following statement as either true or false. Let an integer. Then

Elements Of Modern Algebra

The figure below is a regular hexagonal pyramid. a Find the lateral area L of the pyramid. b Find the total are...

Elementary Geometry For College Students, 7e

These questions are available for instructors to help assess if you have successfully met the learning objectiv...

Trigonometry (MindTap Course List)

12. Subtract fifteen and eighty-eight ten-thousandths from thirty-six. Express your answer in numerical and wor...

Contemporary Mathematics for Business & Consumers

Finding an Equation of a Tangent Line In Exercises 67-74, (a) find an equation of the tangent line to the graph...

Calculus of a Single Variable

HOW DO YOU SEE IT? Use the figure to fill in the blank with the symbol , , or =. Explain your reasoning. (a) Th...

Calculus: Early Transcendental Functions (MindTap Course List)

Find x such that the line through 2,-3 and 3,2 is perpendicular to the line through -2,4 and x,-1.

Elementary Geometry for College Students

GREAT RECESSION In a survey conducted by AlixPartners of 4980 adults 18 years old and older in June 2009 during...

Finite Mathematics for the Managerial, Life, and Social Sciences

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

Calculus

A series circuit has a resistor with R = 15 Ω, an inductor with L = 2, a capacitor with C = 0.005 F and a 6-V b...

Study Guide for Stewart's Multivariable Calculus, 8th

Finding the Domain and Range of a FunctionIn Exercises 3 ad 4, find the domain and range of the function. f(x,y...

Multivariable Calculus

The absolute maximum value of f(x) = 6x − x2 on [1, 7] is: 3 9 76 −7

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

The following crosstabulation shows household income by educational level of the head of household (Statistical...

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

Applications Use a calculator to solve each problem. Earthquakes An earthquake has a period of 12 second and an...

College Algebra (MindTap Course List)

Find distance y. All dimensions are in millimeters

Mathematics For Machine Technology

Briefly explain what is meant when a researcher reports a significant mean difference between two treatment con...

Research Methods for the Behavioral Sciences (MindTap Course List)

Allegiant Airlines charges a mean base fare of 89. In addition, the airline charges for making a reservation on...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Select a subject and use a full-text database such as PsycARTICLES to locate an empirical journal article that ...

Research Methods for the Behavioral Sciences (MindTap Course List)

A new model of laptop computer can be ordered with one of three screen sizes (10 inches, 12 inches, 15 inches) ...

Introduction To Statistics And Data Analysis

Writing and Solving a Differential Equation In Exercises 23 and 24, write and find the general solution of the ...

Calculus (MindTap Course List)

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