   Chapter 3.3, Problem 38E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object.If the rope makes an angle θ with the plane, then the magnitude of the force is F = μ W μ   sin θ + cos θ where μ is a constant called the coefficient of friction.(a) Find the rate of change of F with respect to θ.(b) When is this rate of change equal to 0?(c) If W = 50 lb and μ = 0.6, draw the graph of F as a function of θ and use it to locate the value of θ for which dF/dθ = 0. Is the value consistent with your answer to part (b)?

(a)

To determine

To find: The rate of change of F with respect to θ. That is,dFdθ.

Explanation

Given:

The magnitute of the force is F=μWμsinθ+cosθ,

Where, μ is a costant and W is weight.

Derivative Rule: Quotient Rule

If f(θ). and g(θ) are both differentiable function, then

ddθ[f(θ)g(θ)]=g(θ)ddθ[f(θ)]f(θ)ddθ[g(θ)][g(θ)]2 (1)

Calculation:

Obtain the derivative of F.

dFdθ=ddθ(F)=ddθ(μWμsinθ+cosθ)

Apply Quotient Rule as shown equation (1),

dFdθ=(μsinθ+cosθ)ddθ

(b)

To determine

To find: The value of θ if the rate of change equal to zero.

(c)

To determine

To draw:. The graph of F as a function of θ and use it to locate the value of θ for which dFdθ=0.

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