Explanation Given: The initial amount invested is $100,000. Withdrawals are made at the rate of $20,000 per year. After 5 years the amount becomes $15,000. Formula used: If money is continuously withdrawn from the account at the rate of $ m per year, then A satisfies the equation d A d t − r A = − m ; where A ( 0 ) = P , d A d t is the rate of change of the amount A , rA is the rate of growth from continuous compounding and m is the rate of withdrawals. Integrating factor: The integrating factor of a function f ( x ) is given by I ( x ) = e ∫ f ( x ) d x . Calculation: Given that the initial amount invested is $100,000. Withdrawals are made at the rate of $20,000 per year. After 5 years the amount becomes $15,000. Then substitute, m = 20 , 000 and P = 100 , 000 in the above equation. Therefore, the amount A at any time t must satisfy d A d t − r A = − 20 , 000 ; A ( 0 ) = 100 , 000 . Now find the integrating factor. I ( t ) = e ∫ − r d t = e − r t Multiply both sides of the equation by the integrating factor. e − r t d A d t − r e − r t A = − 20 , 000 e − r t ( A e − r t ) ′ = − 20 , 000 e − r t [ ∵ ( x y ) ′ = x y ′ + x ′ y ] Integrating on both sides, ∫ ( A e − r t ) ′ d t = ∫ − 20 , 000 e − r t d t A e − r t = − 20 , 000 ( e − r t − r ) + C A e − r t = 20 , 000 r e − r t + C A = 20 , 000 r + C e r t The initial condition is A ( 0 ) = 100 , 000

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

14th Edition

ISBN: 9780134668574

Author: Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Christopher J. Stocker

Publisher: PEARSON

See similar textbooks

Concept explainers

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 …

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experime…

Videos

Question

Chapter 9.3, Problem 66E

To determine

To find: The continuous compound rate of interest that has earned over 5-year period.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.3, Problem 65E

Chapter 9.3, Problem 67E

Students have asked these similar questions

10. Prove that, if (t)=1+0(12) as asf->> O is a characteristic function, then p = 1.

9. The concentration function of a random variable X is defined as Qx(h) sup P(x ≤x≤x+h), h>0. (b) Is it true that Qx(ah) =aQx (h)?

3. Let X1, X2,..., X, be independent, Exp(1)-distributed random variables, and set V₁₁ = max Xk and W₁ = X₁+x+x+ Isk≤n

Chapter 9 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

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 continuous compound rate of interest that has earned over 5-year period.

Solution Summary: The author calculates the continuous compound rate of interest that has earned over a 5-year period. The integrating factor is given by I(x)=edisplaystyle

Concept explainers

Videos

To find: The continuous compound rate of interest that has earned over 5-year period.

Want to see the full answer?

Chapter 9 Solutions