Suppose that Po is invested in a savings account in which interest is compounded continuously at 6.1% per year. That is, the balance P grows at the rate given by the following equation. dP = 0.061P(t) dt (a)Find the function P(t) that satisfies the equation. Write it in terms of Po and 0.061. (b)Suppose that $500 is invested. What is the balance after 3 years? (c)When will an investment of $500 double itself?
Suppose that Po is invested in a savings account in which interest is compounded continuously at 6.1% per year. That is, the balance P grows at the rate given by the following equation. dP = 0.061P(t) dt (a)Find the function P(t) that satisfies the equation. Write it in terms of Po and 0.061. (b)Suppose that $500 is invested. What is the balance after 3 years? (c)When will an investment of $500 double itself?
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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![### Continuous Compound Interest Calculations
**Problem Statement:**
Suppose that \( P_0 \) is invested in a savings account in which interest is compounded continuously at 6.1% per year. That is, the balance \( P \) grows at the rate given by the following equation.
\[ \frac{dP}{dt} = 0.061P(t) \]
**Tasks:**
1. Find the function \( P(t) \) that satisfies the equation. Write it in terms of \( P_0 \) and 0.061.
2. Suppose that $500 is invested. What is the balance after 3 years?
3. When will an investment of $500 double itself?
### (a) Find the Function \( P(t) \)
Choose the correct answer below:
- **A.** \( P(t) = 0.061P_0 e^t \)
- **B.** \( P(t) = P(t)e^{0.061t} \)
- **C.** \( P_0 = P(t)e^{0.061t} \)
- **D.** \( \mathbf{P(t) = P_0 e^{0.061t}} \) (Correct Answer)
### (b) Balance After 3 Years
Suppose $500 is invested. What is the balance after 3 years?
\[ \text{The balance after 3 year is } \$ \_\_\_\_. \]
(Type an integer or decimal rounded to two decimal places as needed.)
### (c) Doubling Time
When will an investment of $500 double itself?
\[ \text{The doubling time is } \_\_\_\_ \text{ year(s)}. \]
(Type an integer or decimal rounded to two decimal places as needed.)
---
**Explanation of the Equation:**
- The differential equation \(\frac{dP}{dt} = 0.061P(t)\) suggests a continuously compounded interest problem.
- The solution to this differential equation is given by the function \( P(t) = P_0 e^{0.061t} \).
- Here, \( P_0 \) is the initial amount invested, and \( t \) represents time in years. The term \( 0.061 \) is the interest rate expressed as a decimal.
**Example Calculation:**
1. **Balance After 3 Years:**
\[ P(t)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12c5954d-a211-4538-ad4a-fb5368695090%2F67aa8768-e210-4774-b5d6-64da2b180db8%2Fyrbi5he.png&w=3840&q=75)
Transcribed Image Text:### Continuous Compound Interest Calculations
**Problem Statement:**
Suppose that \( P_0 \) is invested in a savings account in which interest is compounded continuously at 6.1% per year. That is, the balance \( P \) grows at the rate given by the following equation.
\[ \frac{dP}{dt} = 0.061P(t) \]
**Tasks:**
1. Find the function \( P(t) \) that satisfies the equation. Write it in terms of \( P_0 \) and 0.061.
2. Suppose that $500 is invested. What is the balance after 3 years?
3. When will an investment of $500 double itself?
### (a) Find the Function \( P(t) \)
Choose the correct answer below:
- **A.** \( P(t) = 0.061P_0 e^t \)
- **B.** \( P(t) = P(t)e^{0.061t} \)
- **C.** \( P_0 = P(t)e^{0.061t} \)
- **D.** \( \mathbf{P(t) = P_0 e^{0.061t}} \) (Correct Answer)
### (b) Balance After 3 Years
Suppose $500 is invested. What is the balance after 3 years?
\[ \text{The balance after 3 year is } \$ \_\_\_\_. \]
(Type an integer or decimal rounded to two decimal places as needed.)
### (c) Doubling Time
When will an investment of $500 double itself?
\[ \text{The doubling time is } \_\_\_\_ \text{ year(s)}. \]
(Type an integer or decimal rounded to two decimal places as needed.)
---
**Explanation of the Equation:**
- The differential equation \(\frac{dP}{dt} = 0.061P(t)\) suggests a continuously compounded interest problem.
- The solution to this differential equation is given by the function \( P(t) = P_0 e^{0.061t} \).
- Here, \( P_0 \) is the initial amount invested, and \( t \) represents time in years. The term \( 0.061 \) is the interest rate expressed as a decimal.
**Example Calculation:**
1. **Balance After 3 Years:**
\[ P(t)
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