If you invest $200 into a savings account that earms 4% annual interest compounded continuously, how much money will you have after 15 years? Round to TWO decimal places (because its money) Type your answer.
If you invest $200 into a savings account that earms 4% annual interest compounded continuously, how much money will you have after 15 years? Round to TWO decimal places (because its money) Type your answer.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Continuous Compound Interest Calculation Example**
In this example, we will explore the concept of continuous compound interest through a practical scenario:
**Problem:**
If you invest $200 into a savings account that earns a 4% annual interest rate, compounded continuously, how much money will you have after 15 years? Please round your answer to two decimal places, as this represents money.
**Solution Approach:**
To solve this problem, we will use the formula for continuous compound interest:
\[ A = Pe^{rt} \]
Where:
- \( A \) is the amount of money accumulated after time \( t \),
- \( P \) is the principal amount ($200 in this case),
- \( r \) is the annual interest rate (4% or 0.04 as a decimal),
- \( t \) is the time the money is invested for in years (15 years),
- \( e \) is the base of the natural logarithm (approximately 2.71828).
**Calculation Steps:**
1. Convert the interest rate from a percentage to a decimal:
\[ r = 4\% = 0.04 \]
2. Substitute the values into the formula:
\[ A = 200 \times e^{(0.04 \times 15)} \]
3. Compute the exponent:
\[ e^{(0.04 \times 15)} = e^{0.6} \]
4. Calculate the final accumulated amount using a calculator:
\[ A \approx 200 \times 1.82212 = 364.42 \]
**Result:**
After 15 years, your investment will grow to approximately $364.42.
This example illustrates how continuous compounding can significantly increase the value of an investment over time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ad9e91f-22eb-4b2e-aa45-04dba37a70d9%2F9a033668-f6f2-48e2-abf5-fc89daed324d%2Fevbiws_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Continuous Compound Interest Calculation Example**
In this example, we will explore the concept of continuous compound interest through a practical scenario:
**Problem:**
If you invest $200 into a savings account that earns a 4% annual interest rate, compounded continuously, how much money will you have after 15 years? Please round your answer to two decimal places, as this represents money.
**Solution Approach:**
To solve this problem, we will use the formula for continuous compound interest:
\[ A = Pe^{rt} \]
Where:
- \( A \) is the amount of money accumulated after time \( t \),
- \( P \) is the principal amount ($200 in this case),
- \( r \) is the annual interest rate (4% or 0.04 as a decimal),
- \( t \) is the time the money is invested for in years (15 years),
- \( e \) is the base of the natural logarithm (approximately 2.71828).
**Calculation Steps:**
1. Convert the interest rate from a percentage to a decimal:
\[ r = 4\% = 0.04 \]
2. Substitute the values into the formula:
\[ A = 200 \times e^{(0.04 \times 15)} \]
3. Compute the exponent:
\[ e^{(0.04 \times 15)} = e^{0.6} \]
4. Calculate the final accumulated amount using a calculator:
\[ A \approx 200 \times 1.82212 = 364.42 \]
**Result:**
After 15 years, your investment will grow to approximately $364.42.
This example illustrates how continuous compounding can significantly increase the value of an investment over time.
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