Single Variable Calculus: Concepts and Contexts, Enhanced Edition
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
4th Edition
ISBN: 9781337687805
Author: James Stewart
Publisher: Cengage Learning
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Chapter 4.6, Problem 26E
To determine

To calculate: The length of the shortest ladder that will reach from the ground over the fence to the well of the building. If fence 8ft tell runs parallel to a tall building at a distance of 4ft from the building.

Expert Solution & Answer
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Answer to Problem 26E

The length of the shortest ladder that will reach from the ground over the fence to the well of the building 16.65ft

Explanation of Solution

Given information:

A fence 8ft tell runs parallel to a tall building at a distance of 4ft from the building.

.

Formula used:

Pythagorean Theorem: The sum of the squares on the legs of the right angled triangle is equal to the square on the side opposite to the right angle triangle. That is:

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 4.6, Problem 26E , additional homework tip  1

  (H)2=(P)2+(B)2

Similarity property: If two triangles say ABCand PQR are similar. Then

Corresponding sides of the triangles are in proportional, i.e

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 4.6, Problem 26E , additional homework tip  2

  ABPQ=ACPR=ABQR

And

Let f be a differentiable function defined on an interval I and let aI .

Then

  1. x=a is a point of local maximum value of f, if
    1. f(a)=0 and
    2. f(x) changes sign from positive to negative as x increases through a , i.e. if f(x)>0 at every point sufficiently close to and to the left of a , and f(x)<0 at every point sufficiently close to and to the right of a , then a is a point of local maxima
  2. x=a is a point of local maximum value of f, if
    1. f(a)=0 and
    2. f(x) changes sign from negative to positive as x increases through a , i.e. if f(x)<0 at every point sufficiently close to and to the left of a , and at f(x)>0 every point sufficiently close to and to the right of a , then a is a point of local minima.
  3. f(a)=0 and If f(x) does not change sign as x increases through a , then a is neither a point of local maxima nor a point of local minima.
    • If f(a)>0 then f has a local minimum at x=a
    • If f(a)<0 then f has a local maximum at x=a

Calculation:

As per the given problem

Draw the diagram of a fence 8ft tell runs parallel to a tall building at a distance of 4ft from the building

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 4.6, Problem 26E , additional homework tip  3

Recall that, If two triangles say ABCand PQR are similar. Then

Corresponding sides of the triangles are in proportional,

Here, ADEand ABC are similar,

Therefore,

  h8=x+4xh=8(x+4)x

Recall that, Pythagorean Theorem: The sum of the squares on the legs of the right angled triangle is equal to the square on the side opposite to the right angle triangle.

In ABC

  l2=h2+(x+4)2

Substitute h=8(x+4)x ,and simplified

  l2=h2+(x+4)2l2(x)={8(x+4)x}2+(x+4)2l2(x)={8+32x}2+(x+4)2......(1)

Recall that,

Let f be a differentiable function defined on an interval I and let aI .

Then

  1. x=a is a point of local maximum value of f, if
    1. f(a)=0 and
    2. f(x) changes sign from positive to negative as x increases through a , i.e. if f(x)>0 at every point sufficiently close to and to the left of a , and f(x)<0 at every point sufficiently close to and to the right of a , then a is a point of local maxima
  2. x=a is a point of local maximum value of f, if
    1. f(a)=0 and
    2. f(x) changes sign from negative to positive as x increases through a , i.e. if f(x)<0 at every point sufficiently close to and to the left of a , and at every point sufficiently close to and to the right of a , then a is a point of local minima.
  3. f(a)=0 and If f(x) does not change sign as x increases through a , then a is neither a point of local maxima nor a point of local minima.
    • If f(a)>0 then f has a local minimum at x=a
    • If f(a)<0 then f has a local maximum at x=a

Differentiate on both sides,

  ddxl2(x)=ddx[(8+32x)2+(x+4)2]=2(8+32x)×32x2+2(x+4)×1=64(32+8xx3)+2(x+4)......(2)

Solve for ddxl2(x)=0 , and simplified

  64(32+8xx3)+2(x+4)=064×8(4+xx3)=2(x+4)1x3=1256x3=256

Take cube root on both sides,

  x=2563x6.35

Differentiate equation (2) with respect to l

  d2dx2l2(x)=64ddx(32+8xx3)+2ddx(x+4)=64(96x416x2)+2=64(96x4+16x2)+2

Therefore,

  d2dx2l2(x)=64(96x4+16x2)+20

For x6.35 the length of ladder is minimum

Substitute x6.35 in equation (1) and simplified

  l2(6.35)={8+326.35}2+(6.35+4)2169.78+107.12276.90

Take square root on both sides,

  l(6.35)16.65

Conclusion:

Thus the length of the shortest ladder that will reach from the ground over the fence to the well of the building 16.65ft

Chapter 4 Solutions

Single Variable Calculus: Concepts and Contexts, Enhanced Edition

Ch. 4.1 - Prob. 11ECh. 4.1 - Prob. 12ECh. 4.1 - Prob. 13ECh. 4.1 - Prob. 14ECh. 4.1 - Prob. 15ECh. 4.1 - Prob. 16ECh. 4.1 - Prob. 17ECh. 4.1 - Prob. 18ECh. 4.1 - Prob. 19ECh. 4.1 - Prob. 20ECh. 4.1 - Prob. 21ECh. 4.1 - Prob. 22ECh. 4.1 - Prob. 23ECh. 4.1 - Prob. 24ECh. 4.1 - Prob. 25ECh. 4.1 - Prob. 26ECh. 4.1 - Prob. 27ECh. 4.1 - Prob. 28ECh. 4.1 - Prob. 29ECh. 4.1 - Prob. 30ECh. 4.1 - Prob. 31ECh. 4.1 - Prob. 32ECh. 4.1 - Prob. 33ECh. 4.1 - Prob. 34ECh. 4.1 - Prob. 35ECh. 4.1 - Prob. 36ECh. 4.1 - Prob. 37ECh. 4.1 - Prob. 38ECh. 4.1 - Prob. 39ECh. 4.1 - Prob. 40ECh. 4.1 - Prob. 41ECh. 4.1 - Prob. 42ECh. 4.1 - Prob. 43ECh. 4.1 - Prob. 44ECh. 4.2 - Explain the difference between an absolute minimum...Ch. 4.2 - Prob. 2ECh. 4.2 - Prob. 3ECh. 4.2 - For each of the numbers a, b, c, d, r, and s,...Ch. 4.2 - Prob. 5ECh. 4.2 - Use the graph to state the absolute and local...Ch. 4.2 - Prob. 7ECh. 4.2 - Prob. 8ECh. 4.2 - Prob. 9ECh. 4.2 - Prob. 10ECh. 4.2 - (a) Sketch the graph of a function that has a...Ch. 4.2 - Prob. 12ECh. 4.2 - (a) Sketch the graph of a function on [1, 2] that...Ch. 4.2 - Prob. 14ECh. 4.2 - Prob. 15ECh. 4.2 - Prob. 16ECh. 4.2 - Prob. 17ECh. 4.2 - Prob. 18ECh. 4.2 - Prob. 19ECh. 4.2 - Prob. 20ECh. 4.2 - Prob. 21ECh. 4.2 - Prob. 22ECh. 4.2 - Prob. 23ECh. 4.2 - Find the critical numbers of the function. f(x) =...Ch. 4.2 - Find the critical numbers of the function. f(x) =...Ch. 4.2 - Prob. 26ECh. 4.2 - Find the critical numbers of the function. g(t) =...Ch. 4.2 - Prob. 28ECh. 4.2 - Find the critical numbers of the function....Ch. 4.2 - Prob. 30ECh. 4.2 - Prob. 31ECh. 4.2 - Prob. 32ECh. 4.2 - Prob. 33ECh. 4.2 - Find the critical numbers of the function. g() = 4...Ch. 4.2 - Find the critical numbers of the function. f() = 2...Ch. 4.2 - Find the critical numbers of the function. h(t) =...Ch. 4.2 - Find the critical numbers of the function. f(x) =...Ch. 4.2 - Prob. 38ECh. 4.2 - Prob. 39ECh. 4.2 - A formula for the derivative of a function f is...Ch. 4.2 - Prob. 41ECh. 4.2 - Prob. 42ECh. 4.2 - 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Prob. 13ECh. 4.4 - Prob. 14ECh. 4.4 - Prob. 15ECh. 4.4 - Prob. 16ECh. 4.4 - Prob. 17ECh. 4.4 - Prob. 18ECh. 4.4 - Prob. 19ECh. 4.4 - Prob. 20ECh. 4.4 - Prob. 21ECh. 4.4 - Prob. 22ECh. 4.4 - Prob. 23ECh. 4.4 - Prob. 24ECh. 4.4 - Prob. 25ECh. 4.4 - Prob. 26ECh. 4.4 - Prob. 27ECh. 4.4 - Prob. 28ECh. 4.4 - Prob. 29ECh. 4.4 - Prob. 30ECh. 4.4 - Prob. 31ECh. 4.4 - Prob. 32ECh. 4.4 - Prob. 33ECh. 4.4 - Prob. 34ECh. 4.4 - Prob. 35ECh. 4.4 - Prob. 36ECh. 4.5 - Given that...Ch. 4.5 - Given that...Ch. 4.5 - Prob. 3ECh. 4.5 - Given that...Ch. 4.5 - Prob. 5ECh. 4.5 - Prob. 6ECh. 4.5 - Prob. 7ECh. 4.5 - Prob. 8ECh. 4.5 - Prob. 9ECh. 4.5 - Prob. 10ECh. 4.5 - Prob. 11ECh. 4.5 - Prob. 12ECh. 4.5 - Prob. 13ECh. 4.5 - Prob. 14ECh. 4.5 - Prob. 15ECh. 4.5 - Prob. 16ECh. 4.5 - Prob. 17ECh. 4.5 - Prob. 18ECh. 4.5 - Prob. 19ECh. 4.5 - Prob. 20ECh. 4.5 - Prob. 21ECh. 4.5 - Prob. 22ECh. 4.5 - Prob. 23ECh. 4.5 - Prob. 24ECh. 4.5 - Prob. 25ECh. 4.5 - Prob. 26ECh. 4.5 - Prob. 27ECh. 4.5 - Prob. 28ECh. 4.5 - Prob. 29ECh. 4.5 - 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Find two positive numbers whose product is 100 and...Ch. 4.6 - The sum of two positive numbers is 16. What is the...Ch. 4.6 - Prob. 5ECh. 4.6 - Prob. 6ECh. 4.6 - Prob. 7ECh. 4.6 - The rate (in mg carbon/m3/h) at which...Ch. 4.6 - Consider the following problem: A farmer with 750...Ch. 4.6 - Prob. 10ECh. 4.6 - Prob. 11ECh. 4.6 - Prob. 12ECh. 4.6 - Prob. 13ECh. 4.6 - Prob. 14ECh. 4.6 - Prob. 15ECh. 4.6 - Prob. 16ECh. 4.6 - Prob. 17ECh. 4.6 - Prob. 18ECh. 4.6 - Prob. 19ECh. 4.6 - Prob. 20ECh. 4.6 - Prob. 21ECh. 4.6 - Prob. 22ECh. 4.6 - Prob. 23ECh. 4.6 - Prob. 24ECh. 4.6 - Prob. 25ECh. 4.6 - Prob. 26ECh. 4.6 - Prob. 27ECh. 4.6 - Prob. 28ECh. 4.6 - Prob. 29ECh. 4.6 - Prob. 30ECh. 4.6 - Prob. 31ECh. 4.6 - Prob. 32ECh. 4.6 - Prob. 33ECh. 4.6 - Prob. 34ECh. 4.6 - Prob. 35ECh. 4.6 - Prob. 36ECh. 4.6 - Prob. 37ECh. 4.6 - Prob. 38ECh. 4.6 - Prob. 39ECh. 4.6 - Prob. 40ECh. 4.6 - Prob. 41ECh. 4.6 - Prob. 42ECh. 4.6 - Prob. 43ECh. 4.6 - Prob. 44ECh. 4.6 - Prob. 45ECh. 4.6 - Prob. 46ECh. 4.6 - Prob. 47ECh. 4.6 - Prob. 48ECh. 4.6 - Prob. 49ECh. 4.6 - Prob. 50ECh. 4.6 - Prob. 51ECh. 4.6 - Prob. 52ECh. 4.6 - Prob. 53ECh. 4.6 - Prob. 54ECh. 4.6 - Prob. 55ECh. 4.6 - Prob. 56ECh. 4.6 - Prob. 57ECh. 4.6 - Prob. 58ECh. 4.6 - Prob. 59ECh. 4.6 - Prob. 60ECh. 4.6 - Prob. 61ECh. 4.6 - Prob. 62ECh. 4.7 - The figure shows the graph of a function f....Ch. 4.7 - Follow the instructions for Exercise 1(a) but use...Ch. 4.7 - Suppose the tangent line to the curve y = f(x) at...Ch. 4.7 - For each initial approximation, determine...Ch. 4.7 - Prob. 5ECh. 4.7 - Prob. 6ECh. 4.7 - Prob. 7ECh. 4.7 - Prob. 8ECh. 4.7 - Use Newtons method with initial approximation x1 =...Ch. 4.7 - Use Newtons method with initial approximation x1 =...Ch. 4.7 - Prob. 11ECh. 4.7 - Prob. 12ECh. 4.7 - Prob. 13ECh. 4.7 - Prob. 14ECh. 4.7 - Prob. 15ECh. 4.7 - Prob. 16ECh. 4.7 - Prob. 17ECh. 4.7 - Prob. 18ECh. 4.7 - Prob. 19ECh. 4.7 - Prob. 20ECh. 4.7 - Prob. 21ECh. 4.7 - Prob. 22ECh. 4.7 - (a) Apply Newtons method to the equation x2 a = 0...Ch. 4.7 - (a) Apply Newtons method to the equation 1/x a =...Ch. 4.7 - (a) Use Newtons method with x1 = 1 to find the...Ch. 4.7 - Explain why Newtons method fails when applied to...Ch. 4.7 - Prob. 28ECh. 4.7 - Prob. 29ECh. 4.7 - Prob. 30ECh. 4.7 - Prob. 31ECh. 4.7 - Prob. 32ECh. 4.7 - Prob. 33ECh. 4.7 - Prob. 34ECh. 4.8 - Prob. 1ECh. 4.8 - Prob. 2ECh. 4.8 - Prob. 3ECh. 4.8 - Prob. 4ECh. 4.8 - Prob. 5ECh. 4.8 - Prob. 6ECh. 4.8 - Prob. 7ECh. 4.8 - Prob. 8ECh. 4.8 - Prob. 9ECh. 4.8 - Prob. 10ECh. 4.8 - Prob. 11ECh. 4.8 - Prob. 12ECh. 4.8 - Prob. 13ECh. 4.8 - Prob. 14ECh. 4.8 - Prob. 15ECh. 4.8 - Prob. 16ECh. 4.8 - Prob. 19ECh. 4.8 - Prob. 20ECh. 4.8 - Prob. 21ECh. 4.8 - Prob. 22ECh. 4.8 - Prob. 23ECh. 4.8 - Prob. 24ECh. 4.8 - Prob. 25ECh. 4.8 - Prob. 26ECh. 4.8 - Prob. 27ECh. 4.8 - Prob. 28ECh. 4.8 - Prob. 29ECh. 4.8 - Prob. 30ECh. 4.8 - Prob. 31ECh. 4.8 - Prob. 32ECh. 4.8 - Prob. 33ECh. 4.8 - Prob. 34ECh. 4.8 - Prob. 35ECh. 4.8 - Prob. 36ECh. 4.8 - Prob. 37ECh. 4.8 - Prob. 38ECh. 4.8 - The graph of f is shown in the figure. Sketch the...Ch. 4.8 - Prob. 40ECh. 4.8 - Prob. 41ECh. 4.8 - Prob. 42ECh. 4.8 - Prob. 43ECh. 4.8 - Prob. 44ECh. 4.8 - Prob. 45ECh. 4.8 - Prob. 46ECh. 4.8 - Prob. 47ECh. 4.8 - Prob. 48ECh. 4.8 - Prob. 49ECh. 4.8 - Prob. 50ECh. 4.8 - Prob. 51ECh. 4.8 - Prob. 52ECh. 4.8 - Prob. 53ECh. 4.8 - Prob. 54ECh. 4.8 - Prob. 55ECh. 4.8 - Prob. 56ECh. 4.8 - Prob. 57ECh. 4.8 - Prob. 58ECh. 4 - Prob. 1RCCCh. 4 - Prob. 2RCCCh. 4 - Prob. 3RCCCh. 4 - Prob. 4RCCCh. 4 - Prob. 5RCCCh. 4 - Prob. 6RCCCh. 4 - Prob. 7RCCCh. 4 - Prob. 8RCCCh. 4 - Prob. 9RCCCh. 4 - Prob. 10RCCCh. 4 - Prob. 1RQCh. 4 - Prob. 2RQCh. 4 - Prob. 3RQCh. 4 - Prob. 4RQCh. 4 - Prob. 5RQCh. 4 - Prob. 6RQCh. 4 - Prob. 7RQCh. 4 - Prob. 8RQCh. 4 - Prob. 9RQCh. 4 - Prob. 10RQCh. 4 - Prob. 11RQCh. 4 - Prob. 12RQCh. 4 - Prob. 13RQCh. 4 - If f and g are positive increasing functions on an...Ch. 4 - Prob. 15RQCh. 4 - Prob. 16RQCh. 4 - Prob. 17RQCh. 4 - Prob. 18RQCh. 4 - If f(x) exists and is nonzero for all x, then f(1)...Ch. 4 - limx0xex=1Ch. 4 - Prob. 1RECh. 4 - Prob. 2RECh. 4 - Prob. 3RECh. 4 - Prob. 4RECh. 4 - Prob. 5RECh. 4 - Prob. 6RECh. 4 - Prob. 7RECh. 4 - Prob. 8RECh. 4 - Prob. 9RECh. 4 - Prob. 10RECh. 4 - Prob. 11RECh. 4 - Prob. 12RECh. 4 - Prob. 13RECh. 4 - Prob. 14RECh. 4 - Prob. 15RECh. 4 - Prob. 16RECh. 4 - Prob. 17RECh. 4 - Prob. 18RECh. 4 - Prob. 19RECh. 4 - Prob. 20RECh. 4 - Prob. 21RECh. 4 - Prob. 22RECh. 4 - Prob. 23RECh. 4 - Prob. 24RECh. 4 - Prob. 25RECh. 4 - Prob. 26RECh. 4 - Prob. 27RECh. 4 - Prob. 28RECh. 4 - Prob. 29RECh. 4 - Prob. 30RECh. 4 - Prob. 31RECh. 4 - Prob. 32RECh. 4 - Prob. 33RECh. 4 - Prob. 34RECh. 4 - Prob. 35RECh. 4 - Prob. 36RECh. 4 - Prob. 37RECh. 4 - Prob. 38RECh. 4 - Prob. 39RECh. 4 - Prob. 40RECh. 4 - Prob. 41RECh. 4 - Prob. 42RECh. 4 - Prob. 43RECh. 4 - Prob. 44RECh. 4 - Prob. 45RECh. 4 - Prob. 46RECh. 4 - Prob. 47RECh. 4 - Prob. 48RECh. 4 - Prob. 49RECh. 4 - Prob. 50RECh. 4 - Prob. 51RECh. 4 - Prob. 52RECh. 4 - Prob. 53RECh. 4 - Prob. 54RECh. 4 - Prob. 55RECh. 4 - Prob. 56RECh. 4 - Prob. 57RECh. 4 - Prob. 58RECh. 4 - Prob. 59RECh. 4 - Prob. 60RECh. 4 - Prob. 61RECh. 4 - Prob. 62RECh. 4 - Prob. 63RECh. 4 - Prob. 64RECh. 4 - Prob. 65RECh. 4 - If a rectangle has its base on the x-axis and two...Ch. 4 - Show that sinxcosx2 for all x.Ch. 4 - Prob. 3PCh. 4 - Prob. 4PCh. 4 - Prob. 5PCh. 4 - Find the point on the parabola y = 1 x2 at which...Ch. 4 - Prob. 7PCh. 4 - Prob. 8PCh. 4 - Prob. 9PCh. 4 - Prob. 10PCh. 4 - Prob. 11PCh. 4 - Prob. 12PCh. 4 - Prob. 13PCh. 4 - Prob. 14PCh. 4 - Prob. 15PCh. 4 - Prob. 16PCh. 4 - Prob. 17PCh. 4 - Prob. 18PCh. 4 - Prob. 19PCh. 4 - Prob. 20PCh. 4 - Prob. 21PCh. 4 - Prob. 22PCh. 4 - Prob. 23PCh. 4 - Prob. 24P
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