Final answer:
To determine how long it will take for an investment of $14,000 to triple at an interest rate of 4.5% compounded continuously, you can use the formula for continuous compound interest. It will take approximately 23 years for the money to triple.
Step-by-step explanation:
To determine how long it will take for an investment of $14,000 to triple at an interest rate of 4.5% compounded continuously, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where A is the final amount, P is the principal (initial investment), e is Euler's number (approximately 2.71828), r is the interest rate, and t is the time in years.
In this case, we want the final amount to be triple the principal, so A = 3P. Substituting the values into the formula, we get:
3P = P * e^(0.045t)
Dividing both sides by P, we can cancel out the P's:
3 = e^(0.045t)
To solve for t, we need to isolate the exponent. Taking the natural logarithm (ln) of both sides:
ln(3) = ln(e^(0.045t))
Using the property of logarithms that ln(e^x) = x, we get:
ln(3) = 0.045t
Finally, dividing both sides by 0.045, we find:
t = ln(3) / 0.045 ≈ 22.47
Therefore, it will take approximately 23 years for the money to triple, rounding up to the nearest year.