(a) Explanation Given: Rate of change in sales S is proportional to L − S where L (in thousands of units) is the estimated maximum level of sales, and S = 0 when t = 0 , for L = 100 , S = 25 when t = 2 Formula used: If a first-order linear differential equation can be expressed in the standard form: d y d x + P ( x ) y = Q ( x ) where P and Q are continuous functions of x . Then, the general solution of the equation can be given as: y = 1 u ( x ) ∫ Q ( x ) u ( x ) d x Where u ( x ) = e ∫ P ( x ) d x Calculation: It is given that rate of change of S is proportional to L − S . Thus, the proportion can be given as: d S d t = k ( L − S ) ……(1) Here S is the sales (in thousands of units) of a new product and L (in thousands of units) is the estimated maximum level of sales. If a first-order linear differential equation can be expressed in the standard form: d y d x + P ( x ) y = Q ( x ) here P and Q are continuous functions of x . Then, the general solution of the equation can be given as: y = 1 u ( x ) ∫ Q ( x ) u ( x ) d x here u ( x ) = e ∫ P ( x ) d x Now, equation (1) can be expressed in standard form by rearranging the terms: d S d t + k S = k L Thus, P ( t ) = k Q ( t ) = k L , u ( t ) = e ∫ P ( t ) d t u ( t ) = e ∫ ( k ) d t u ( t ) = e k t And the solution can be given as: S = 1 u ( t ) ∫ Q ( t ) u ( t ) d t S = 1 e k t ∫ k L ( e k t ) d t S = 1 e t ∫ k L ( e t ) d t S = 1 e k t ( k L e k t k + c ) S = L + c e k t (b) To determine To calculate: Sales S as a function of t where, rate of change in sales S is proportional to L − S where L (in thousands of units) is the estimated maximum level of sales, and S = 0 when t = 0 , for L = 500 , S = 50 when t = 1 .

Bundle: Calculus: Early Transcendental Functions, 7th + Webassign, Multi-term Printed Access Card

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ISBN: 9781337888936

Author: Ron Larson, Bruce H. Edwards

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Chapter 6, Problem 56RE

(a)

To determine

To calculate: Sales S as a function of t where, rate of change in sales S is proportional to L−S where L (in thousands of units) is the estimated maximum level of sales, and S=0 when t=0, for L=100, S=25 when t=2.

(b)

To determine

To calculate: Sales S as a function of t where, rate of change in sales S is proportional to L−S where L (in thousands of units) is the estimated maximum level of sales, and S=0 when t=0, for L=500, S=50 when t=1.

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Chapter 6, Problem 55RE

Chapter 6, Problem 57RE

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Chapter 6 Solutions

Bundle: Calculus: Early Transcendental Functions, 7th + Webassign, Multi-term Printed Access Card

Ch. 6.1 - Verifying a Solution Describe how to determine...Ch. 6.1 - General Solution What does the general solution of...Ch. 6.1 - Slope Field What do the line segments on a slope...Ch. 6.1 - Euler's Method What does Eulers Method allow you...Ch. 6.1 - Verify that the function y=Ce5x is a solution of...Ch. 6.1 - Verify that the function y=e2x is a solution of...Ch. 6.1 - Verify that the function y=C1sinxC2cosx is a...Ch. 6.1 - Verify that the function y=C1excosx+C2exsinx is a...Ch. 6.1 - Verify that the function y=(cosx)lnsecx+tanx is a...Ch. 6.1 - Verify that the function y=25(e4x+ex) is a...

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Sales S as a function of t where, rate of change in sales S is proportional to L − S where L (in thousands of units) is the estimated maximum level of sales, and S = 0 when t = 0 , for L = 100 , S = 25 when t = 2 .

Solution Summary: The author explains that rate of change in sales is proportional to L-S where L (in thousands of units) is the estimated maximum level of sales.

To calculate: Sales S as a function of t where, rate of change in sales S is proportional to L−S where L (in thousands of units) is the estimated maximum level of sales, and S=0 when t=0, for L=100, S=25 when t=2.

To calculate: Sales S as a function of t where, rate of change in sales S is proportional to L−S where L (in thousands of units) is the estimated maximum level of sales, and S=0 when t=0, for L=500, S=50 when t=1.

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Chapter 6 Solutions