Sure! Let's solve this problem step-by-step.
### Step 1: Understand the Problem
We need to calculate the future value of an initial principal amount of [tex]$100,000 that is compounded continuously at an annual interest rate of 3.5% over 20 years. We then need to round this future value to the nearest dollar.
### Step 2: Identify the Formula
For continuous compounding, the formula to calculate the future value (A) is:
\[ A = P \times e^{(rt)} \]
where:
- \( P \) is the principal amount (initial investment)
- \( r \) is the annual interest rate (expressed as a decimal)
- \( t \) is the time in years
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828)
### Step 3: Plug in the Values
Given:
- \( P = 100,000 \)
- \( r = 0.035 \) (3.5% annual interest rate)
- \( t = 20 \) years
### Step 4: Calculate the Accumulated Value
We use the formula:
\[ A = 100,000 \times e^{(0.035 \times 20)} \]
### Step 5: Evaluate the Exponential Component
First, compute the exponent:
\[ 0.035 \times 20 = 0.7 \]
Then, compute \( e^{0.7} \):
\[ e^{0.7} \approx 2.01375 \]
### Step 6: Multiply by the Principal
Now calculate the future value:
\[ A = 100,000 \times 2.01375 = 201,375.27 \]
### Step 7: Round to the Nearest Dollar
Rounding the accumulated value to the nearest dollar:
\[ \boxed{201,375} \]
### Conclusion
The accumulated future value of a $[/tex]100,000 continuous income stream that has been compounded continuously at 3.5% over 20 years, rounded to the nearest dollar, is $201,375.
