What is the compound interest if $45,000 is invested for 15 years at 8% compounded continuously? (Round your answer to the nearest cent.)

To find the compound interest with continuous compounding, we use the formula:

\[
A = P e^{rt}
\]

where:
- $A$ = future value of the investment/loan, including interest  
- $P$ = principal amount (initial investment)  
- $r$ = annual interest rate (as a decimal)  
- $t$ = time in years  
- $e \approx 2.71828$ is the base of the natural logarithm  

### Step 1: Plug in the values
- $P = 45,000$  
- $r = 0.08$ (8%)  
- $t = 15$ years  

\[
A = 45,000 \times e^{0.08 \times 15}
\]

### Step 2: Simplify the exponent  
\[
0.08 \times 15 = 1.2
\]

So:

\[
A = 45,000 \times e^{1.2}
\]

### Step 3: Evaluate $e^{1.2}$
\[
e^{1.2} \approx 3.32012
\]

Now:

\[
A = 45,000 \times 3.32012 = 149,405.40
\]

### Step 4: Calculate the compound interest  
The compound interest is the difference between the final amount $A$ and the principal $P$.

\[
\text{Compound Interest} = A - P = 149,405.40 - 45,000 = 104,405.40
\]

### Final Answer:
The compound interest is **$104,405.40**.

Would you like further details or explanations?

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Here are 5 related questions to explore:  
1. What is the formula for compound interest with annual compounding?  
2. How would the interest change if the rate was compounded monthly?  
3. What happens to the final amount if the interest rate is doubled?  
4. How does continuous compounding differ from simple interest?  
5. What is the meaning of the constant $e$ in mathematics?

**Tip:** Continuous compounding maximizes growth since interest is calculated at every moment rather than in discrete intervals.

What is the compound interest if $45,000 is invested for 15 years at 8% compounded continuously? (Round your answer to the nearest cent.)

Explore how to calculate compound interest on a $45,000 investment over 15 years at an 8% rate compounded continuously. Learn the exponential growth formula for continuous compounding with step-by-step calculations, resulting in a total compound interest of $104,405.40. Suitable for students in Grades 11-12.

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