(A) Explanation Theorem used: Separation of Variables: The solution of the differential equation f ( y ) y ′ = g ( x ) is given implicitly by the equation ∫ f ( y ) d y = ∫ g ( x ) d x . Formula used: Limited growth model: “If the rate of change with respect to time t of a quantity y is proportional to the difference between y and a limiting value M , then y satisfies the differential equation d y d t = k ( M − y ) , k > 0 , t > 0 .” Calculation: Let S represents the annual sales of the company with the initial condition S ( 0 ) = 0 . In first year, the sale of the company is $50,000. The rate of growth of annual sales of a company is proportional to the difference between annual sales S and $200,000. This implies, d S d t = k ( 200 , 000 − S ) , k > 0 , t > 0 Thus, the problem follows a limited growth model. Rewrite the differential equation d S d t = k ( 200 , 000 − S ) in the form of equation f ( y ) y ′ = g ( x ) as follows. d S ( 200 , 000 − S ) = k d t 1 ( 200 , 000 − S ) d S = k d t Compare the equation 1 ( 200 , 000 − S ) d S = k d t with equation f ( y ) y ′ = g ( x ) . It is observed that, f ( S ) = 1 ( 200 , 000 − S ) g ( t ) = k By the theorem, the solution is given by, ∫ f ( S ) d S = ∫ g ( t ) d t ∫ ( 1 200 , 000 − S ) d S = ∫ k d t − ln ( 200 , 000 − S ) = k t + C 1 ln ( 200 , 000 − S ) − 1 = k t + C 1 Simplify further as, e ln ( 200 , 000 − S ) − 1 = e k t + C 1 ( 200 , 000 − S ) − 1 = e k t + C 1 200 , 000 − S = e − k t − C 1 S = 200 , 000 − C e − k t , where C = e − C 1 Given that, S ( 0 ) = 0 . Substitute t = 0 and S = 0 in S = 200 , 000 − C e − k t and obtain the value of C (B) To determine To find: The sales of first year if the sales must to be achieve $150,000 after 3 years.

Pearson eText for Calculus for Business, Economics, Life Sciences, and Social Sciences, Brief Version -- Instant Access (Pearson+)

14th Edition

ISBN: 9780137400126

Author: Raymond Barnett, Michael Ziegler

Publisher: PEARSON+

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 9, Problem 41RE

(A)

To determine

To find: The time required to reach the annual sale of $150,000.

(B)

To determine

To find: The sales of first year if the sales must to be achieve $150,000 after 3 years.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9, Problem 40RE

Chapter 9, Problem 42RE

Students have asked these similar questions

Title: Analyzing Customer Satisfaction for UnileverAs a member of Unilever's Customer Experience Management team, you are responsible forevaluating customer satisfaction levels and monitoring competitive moves. This case studyinvolves analyzing satisfaction data to test two key hypotheses about Unilever's performancerelative to its main competitor, Procter & Gamble (P&G).Unilever’s leadership team has emphasized the importance of customer satisfaction inmaintaining competitive advantage and market leadership. As part of this initiative, yourteam regularly monitors satisfaction scores and benchmarks them against competitors likeP&G.You are tasked with analyzing the provided dataset to answer the following questions:1. Does Unilever’s average customer satisfaction score meet the minimum threshold of2. 75%?Is there no significant difference between Unilever’s overall average satisfaction scoreand P&G’s average satisfaction score?

Please help me first one graphically and the other in matrix

Please help me with this in matrix please

Chapter 9 Solutions

Pearson eText for Calculus for Business, Economics, Life Sciences, and Social Sciences, Brief Version -- Instant Access (Pearson+)

Ch. 9.1 - Show that y=Cx+1 is the general solution of the...Ch. 9.1 - Prob. 2MP Ch. 9.1 - Prob. 3MP Ch. 9.1 - Prob. 4MP Ch. 9.1 - Prob. 5MP Ch. 9.1 - Prob. 1ED Ch. 9.1 - Prob. 2ED Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The time required to reach the annual sale of $150,000.