Final answer:
To calculate the time it takes for an investment to double when compounded continuously, we use the continuous compound interest formula. For $3250 to grow to $6500 at a 6.5% interest rate, it will take approximately 10.67 years.
Step-by-step explanation:
To determine how long it will take for $3250 to grow to $6500 at a continuous compounding interest rate of 6.5%, we can use the formula for continuous compound interest:
A = Pert
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- t is the time the money is invested for, in years.
- e is the base of the natural logarithm, approximately equal to 2.71828.
We can rearrange the formula to solve for t:
t = (ln(A/P)) / r
Plug in the values (A = $6500, P = $3250, and r = 0.065):
t = (ln(6500/3250)) / 0.065
Calculate the natural logarithm and divide by the interest rate:
t = (ln(2)) / 0.065 ≈ 10.67 years
Therefore, it will take approximately 10.67 years for the balance to reach $6500 when compounded continuously at a 6.5% interest rate.