You want to make an investment in a continuously compounding account over a period of 20 years. What interest rate is required for your investment to double in that time period? Round the logarithm value to the nearest hundredth and the answer to the nearest tenth.
To find the required interest rate for your investment to double in a continuously compounding account over a period of 20 years, we can use the formula for compound interest:
A = P * e^(rt)
Where:
A = Final amount (double the initial investment)
P = Principal amount (initial investment)
e = Euler's number (approximately 2.71828)
r = Interest rate (unknown)
t = Time period (20 years)
We can rearrange the formula to solve for the interest rate (r):
r = ln(A/P) / t
Since we want the investment to double, the final amount (A) is equal to 2 times the initial investment (P):
r = ln(2P/P) / 20
Simplifying the expression further:
r = ln(2) / 20
Using a scientific calculator or a calculator with logarithmic functions, we can find the natural logarithm of 2 (ln(2)). The approximate value of ln(2) is 0.69314 when rounded to five decimal places.
r ≈ 0.69314 / 20
r ≈ 0.03466
Rounding the interest rate (r) to the nearest hundredth, we get approximately 0.03. This means you would need an interest rate of 3% (nearest tenth) for your investment to double in the 20-year time period.