Suppose that Saul has money in a savings account that pays him interest of 5% per year on the account balance. Now, Saul takes out $50 a month for various frivolities, and his mother secretly deposits $240 every 6 months into his savings account. Assuming that interest is paid and money is deposited and withdrawn from the account in a continuous fashion, the balance B = B(t) (in dollars) remaining in Saul's savings account at time t (in months) is best modeled by the differential equation: A. at dB = 0.05B-10 B. dt C. dt 器 D. dt = dB E. dt 0.05B-10 12 0.05B 12 = -- 190 0.05B 12 - 50 = 0.05B-120
Suppose that Saul has money in a savings account that pays him interest of 5% per year on the account balance. Now, Saul takes out $50 a month for various frivolities, and his mother secretly deposits $240 every 6 months into his savings account. Assuming that interest is paid and money is deposited and withdrawn from the account in a continuous fashion, the balance B = B(t) (in dollars) remaining in Saul's savings account at time t (in months) is best modeled by the differential equation: A. at dB = 0.05B-10 B. dt C. dt 器 D. dt = dB E. dt 0.05B-10 12 0.05B 12 = -- 190 0.05B 12 - 50 = 0.05B-120
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Modeling a Savings Account with Continuous Interest and Transactions
**Context:**
Suppose that Saul has money in a savings account that pays him an interest of 5% per year on the account balance. Now, Saul takes out $50 a month for various frivolities, and his mother secretly deposits $240 every 6 months into his savings account. Assuming that interest is paid and money is deposited and withdrawn from the account in a continuous fashion, the balance \( B = B(t) \) (in dollars) remaining in Saul's savings account at time \( t \) (in months) is best modeled by the differential equation:
To determine the most accurate model, we are presented with the following options of differential equations:
**Options:**
A. \[ \frac{dB}{dt} = \frac{0.05B}{12} - 10 \]
B. \[ \frac{dB}{dt} = 0.05B - 10 \]
C. \[ \frac{dB}{dt} = \frac{0.05B}{12} - 190 \]
D. \[ \frac{dB}{dt} = \frac{0.05B}{12} - 50 \]
E. \[ \frac{dB}{dt} = 0.05B - 120 \]
### Explanation of Differential Equations:
1. **Interest Calculation:**
- The interest rate of 5% per year is equivalent to a monthly interest rate of \( \frac{0.05}{12} \), since there are 12 months in a year.
2. **Withdrawals:**
- Saul takes out \( $50 \) every month, which affects the balance negatively.
### Examination of Each Option:
- **Option A:**
\[ \frac{dB}{dt} = \frac{0.05B}{12} - 10 \]
- **Interest Component:** \( \frac{0.05B}{12} \), which represents monthly interest.
- **Withdrawal Component:** \( 10 \), representing the conversion of \( $50 \) monthly withdrawals to \( \$ \) per month (divided incorrectly, should consider full 50).
- **Option B:**
\[ \frac{dB}{dt} = 0.05B - 10 \]
- **Interest Component:** \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2bf6a656-c902-430b-8919-3920c6e4552b%2F33dda777-4edf-4a17-b4af-06df344c14f5%2F9oaa7kk_processed.png&w=3840&q=75)
Transcribed Image Text:### Modeling a Savings Account with Continuous Interest and Transactions
**Context:**
Suppose that Saul has money in a savings account that pays him an interest of 5% per year on the account balance. Now, Saul takes out $50 a month for various frivolities, and his mother secretly deposits $240 every 6 months into his savings account. Assuming that interest is paid and money is deposited and withdrawn from the account in a continuous fashion, the balance \( B = B(t) \) (in dollars) remaining in Saul's savings account at time \( t \) (in months) is best modeled by the differential equation:
To determine the most accurate model, we are presented with the following options of differential equations:
**Options:**
A. \[ \frac{dB}{dt} = \frac{0.05B}{12} - 10 \]
B. \[ \frac{dB}{dt} = 0.05B - 10 \]
C. \[ \frac{dB}{dt} = \frac{0.05B}{12} - 190 \]
D. \[ \frac{dB}{dt} = \frac{0.05B}{12} - 50 \]
E. \[ \frac{dB}{dt} = 0.05B - 120 \]
### Explanation of Differential Equations:
1. **Interest Calculation:**
- The interest rate of 5% per year is equivalent to a monthly interest rate of \( \frac{0.05}{12} \), since there are 12 months in a year.
2. **Withdrawals:**
- Saul takes out \( $50 \) every month, which affects the balance negatively.
### Examination of Each Option:
- **Option A:**
\[ \frac{dB}{dt} = \frac{0.05B}{12} - 10 \]
- **Interest Component:** \( \frac{0.05B}{12} \), which represents monthly interest.
- **Withdrawal Component:** \( 10 \), representing the conversion of \( $50 \) monthly withdrawals to \( \$ \) per month (divided incorrectly, should consider full 50).
- **Option B:**
\[ \frac{dB}{dt} = 0.05B - 10 \]
- **Interest Component:** \(
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