Explanation Given: The initial sales of a company were $0 million. The sales were $2 million during first year and $5 million during third year. The annual sales grow at a rate proportional to the difference between the sales and a limit M . 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 . Calculation: The sales is given by the equation S ( t ) = M ( 1 − e − k t ) , M > 0 , k > 0 . The sales were $2 million during first year and $5 million during third year. That is, S ( 1 ) = 2 and S ( 3 ) = 5 . Substitute t = 1 and S = 2 in S ( t ) = M ( 1 − e − k t ) , 2 = M ( 1 − e − k ) M = 2 1 − e − k Substitute M = 2 1 − e − k in S ( t ) = M ( 1 − e − k t ) , S ( t ) = 2 1 − e − k ( 1 − e − k t ) Substitute t = 3 and S = 5 in S ( t ) = 2 1 − e − k ( 1 − e − k t ) . S ( t ) = 2 1 − e − k ( 1 − e − k t ) 5 = 2 1 − e − k ( 1 − e − 3 k ) 5 ( 1 − e − k ) = 2 ( 1 − e − 3 k ) 5 − 5 e − k = 2 − 2 e − 3 k Simplify further as, 2 e − 3 k − 5 e − k + 3 = 0 . Consider t = e − k , then k = − ln t . Thus, the obtained equation becomes, 2 t 3 − 5 t + 3 = 0 . Factorize the cubic polynomial 2 t 3 − 5 t + 3 = 0 . By the trial and error method, suppose t = 1 . 2 ( 1 ) 3 − 5 ( 1 ) + 3 = 2 − 5 + 3 = 0 Thus, t − 1 is a factor of 2 t 3 − 5 t + 3 = 0

14th Edition

ISBN: 9780134856667

Author: Stocker

Publisher: VST

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 9.2, Problem 69E

To determine

To find: The approximate value of k round off to one decimal place and M round off to the nearest million.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.2, Problem 68E

Chapter 9.2, Problem 70E

Students have asked these similar questions

Steel Production Planning: Let S represent the amount of steel produced (in tons). Steel Production is related to the amount of labor used(L) and the amount of capital used ( C ) by the following function:S = 20L⁰˙³⁰C⁰˙⁷⁰In this formula L represents the units of Labor input and C the units of Capital input. Each unit of Labor costs $50, and each unit of Capital costs $100. a:Formulate an optimization problem that will determine how much labor and capital are needed to produce 50,000 tons of steel at minimum cost. Q#3B: Solve the optimization problem you formulated in part (a). (Hint: When using Excel Solver, start with an initial L>0 and C>0.)…

Use the accompanying Grade Point Averages data to find 80%,85%, and 99%confidence intervals for the mean GPA. view the Grade Point Averages data. Gender College GPAFemale Arts and Sciences 3.21Male Engineering 3.87Female Health Science 3.85Male Engineering 3.20Female Nursing 3.40Male Engineering 3.01Female Nursing 3.48Female Nursing 3.26Female Arts and Sciences 3.50Male Engineering 3.00Female Arts and Sciences 3.13Female Nursing 3.34Female Nursing 3.67Female Education 3.45Female Engineering 3.17Female Health Science 3.28Female Nursing 3.25Male Engineering 3.72Female Arts and Sciences 2.68Female Nursing 3.40Female Health Science 3.76Female Arts and Sciences 3.72Female Education 3.44Female Arts and Sciences 3.61Female Education 3.29Female Nursing 3.20Female Education 3.80Female Business 3.26Male…

(3) (20 points) Let F(x, y, z) = (y, z, x²z). Define E = {(x, y, z) | x² + y² ≤ z ≤ 1, x ≤ 0}. (a) (2 points) Calculate the divergence V. F. (b) (4 points) Let D = {(x, y) | x² + y² ≤ 1, x ≤ 0} Without calculation, show that the triple integral √ (V · F) dV = √ 2²(1. = x²(1 − x² - y²) dA. E

Chapter 9 Solutions

EBK CALCULUS FOR BUSINESS, ECONOMICS, L

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

$Calculus$

Learn more about

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

The approximate value of k round off to one decimal place and M round off to the nearest million.

Solution Summary: The author calculates the approximate value of k and M round off to the nearest million.

Concept explainers

Videos

To find: The approximate value of k round off to one decimal place and M round off to the nearest million.

Want to see the full answer?

Chapter 9 Solutions