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# Endowment model An endowment is an investment account in ISBN: 9780321570567 2

## Solution for problem 59E Chapter 7.8

Calculus: Early Transcendentals | 1st Edition

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Problem 59E

Endowment model An endowment is an investment account in which the balance ideally remains constant and withdrawals are made on the interest earned by the account. Such an account may be modeled by the initial value problem B′(t)= aB − m for t ≥ 0, with B(0)= B0. The constant a reflects the annual interest rate, m is the annual rate of withdrawal, and B0, is the initial balance in the account.

a. Solve the initial value problem with a = 0.05, m =\$1000/yr, and B0=\$15,000. Does the balance in the account increase or decrease?

b. If a =0.05 and B0 = \$50,000, what is the annual withdrawal rate m that ensures a constant balance in the account? What is the constant balance?

Step-by-Step Solution:

Problem 59E

Endowment model

An endowment is an investment account in which the balance ideally remains constant and withdrawals are made on the interest earned by the account. Such an account may be modeled by the initial value problem B′(t)= aB − m for t ≥ 0, with B(0)= B0. The constant a reflects the annual interest rate, m is the annual rate of withdrawal, and B0, is the initial balance in the account.

a. Solve the initial value problem with a = 0.05, m =\$1000/yr, and B0=\$15,000. Does the balance in the account increase or decrease?

b. If a =0.05 and B0 = \$50,000, what is the annual withdrawal rate m that ensures a constant balance in the account? What is the constant balance?

Solution:Step 1Solving for initial value problem, Considering , Separating variables, Integrating both sides,  Combining the constants, Using initial condition , So, the final solution becomes,      Putting in the values a = 0.05, m =\$1000/yr, and B0=\$15,000, we get    The balance increases.

Step 2Annual withdrawal rate m that ensures a constant balance in the account is calculated as followsUsing ,Putting the values of   So the annual withdrawal rate m is \$2500/year for ensuring constant balance.

The constant balance is calculated using ,Putting the values for m, a and B0,  Hence the constant balance is \$50,000.

Step 2 of 2

##### ISBN: 9780321570567

Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321570567. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. The full step-by-step solution to problem: 59E from chapter: 7.8 was answered by , our top Calculus solution expert on 03/03/17, 03:45PM. This full solution covers the following key subjects: account, balance, constant, annual, rate. This expansive textbook survival guide covers 85 chapters, and 5218 solutions. Since the solution to 59E from 7.8 chapter was answered, more than 388 students have viewed the full step-by-step answer. The answer to “Endowment model An endowment is an investment account in which the balance ideally remains constant and withdrawals are made on the interest earned by the account. Such an account may be modeled by the initial value problem B?(t)= aB ? m for t ? 0, with B(0)= B0. The constant a reflects the annual interest rate, m is the annual rate of withdrawal, and B0, is the initial balance in the account.a. Solve the initial value problem with a = 0.05, m =\$1000/yr, and B0=\$15,000. Does the balance in the account increase or decrease?________________b. If a =0.05 and B0 = \$50,000, what is the annual withdrawal rate m that ensures a constant balance in the account? What is the constant balance?” is broken down into a number of easy to follow steps, and 121 words.

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