Final answer:
To find the interest rate needed for $25,000 to double after 8 years when compounded continuously quarterly, divide the natural logarithm of 2 by 8 to find the interest rate per period.
Step-by-step explanation:
To find the interest rate needed for $25,000 to double after 8 years when compounded continuously quarterly, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
- A is the final amount
- P is the initial amount
- e is the mathematical constant approximately equal to 2.71828
- r is the interest rate per period
- t is the number of periods
In this case, we want to find the interest rate needed for $25,000 to double, so A = 2P. Plugging in the given values:
2P = P * e^(rt)
Dividing both sides by P:
2 = e^(rt)
Take the natural logarithm of both sides:
ln(2) = rt
Divide both sides by t:
r = ln(2) / t
Substituting the given values:
r = ln(2) / 8
Using a calculator, we find that:
r ≈ 0.0869
Therefore, the interest rate needed for $25,000 to double after 8 years when compounded continuously quarterly is approximately 0.0869, or 8.69%.