Question

irrational

S= Pe

time (years)

10% rate

principal

$10,000

15 yr

1090

645 yr

73.14

1.0875

12

2.7182818

360

Answer 1

It looks like you have a few different concepts related to finance and possibly exponential growth. Let’s break it down based on the pieces of information you provided:

1. **Irrational** - This may refer to a number that cannot be expressed as a simple fraction or it might refer to something else that is not logical or reasonable in a specific context.

2. **S = Pe** - This formula typically represents the future value \( S \) of an investment after time \( t \) years at a continuous growth rate. Specifically:
- \( S \) is the future value of the investment.
- \( P \) is the principal amount (the initial amount of money).
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
- If this is in the context of continuous compounding, we could further specify the formula as:
\[
S = Pe^{rt}
\]
where \( r \) is the annual interest rate (as a decimal) and \( t \) is time in years.

3. **10% rate** - This indicates an interest rate of 10% per annum.

4. **Principal $10,000** - This is the initial investment amount ($10,000).

5. **15 years** - This is the time period over which the investment will grow.

6. **1090** - This value is a bit unclear. It might be a calculation or a specific value relevant to another formula.

7. **645 years** - Similarly, this could refer to the time period or another value of interest.

8. **73.14** - This could be derived from an exponential function or a calculation related to the interest.

9. **1.0875** - This looks like a factor that might represent growth over a specific time frame.

10. **12** - This could refer to the compounding frequency (monthly compounding would be 12 times a year).

11. **2.7182818** - This is the constant \( e\), essential in calculations dealing with exponential growth or decay.

12. **360** - This could refer to financial institutions using a 360-day year in their interest calculations.

### Example Calculation

If we want to calculate the future value of a principal amount of $10,000 at a 10% interest rate for 15 years using \( S = Pe^{rt} \):

Substituting the values:
- \( P = 10,000 \)
- \( r = 0.10 \) (10% as a decimal)
- \( t = 15 \)

The formula becomes:
\[
S = 10,000 \times e^{0.10 \times 15}
\]

Calculating the exponent:
\[
e^{0.10 \times 15} = e^{1.5} \approx 4.4817
\]

Now calculating \( S \):
\[
S = 10,000 \times 4.4817 \approx 44,817
\]

In this scenario, after 15 years, the investment would grow to approximately $44,817 given a 10% annual interest rate with continuous compounding.

If you have a specific question or need to clarify any particular item, please let me know!