Final answer:
To find the time it takes for an investment to double at 2% compounded quarterly, we use the compound interest formula with quarterly compounding parameters to solve for t. For continuous compounding, we use a different formula and solve for t as well. After calculation, the time for an investment to double at 2% compounded quarterly is approximately 35 years and 34.66 years for continuous compounding.
Step-by-step explanation:
To determine how long it takes for an investment to double in value at a 2% interest rate compounded quarterly, we can use the formula for compound interest:
A = P(1 + \frac{r}{n})^{nt}
where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (initial investment).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the number of years the money is invested for.
Since we want the investment to double, A = 2P. Solving for t, we get:
t = \frac{\log(2)}{n \cdot \log(1 + \frac{r}{n})}
Plugging in the values for r = 0.02 (2% interest rate) and n = 4 (compounded quarterly), we can solve for t.
To calculate the time to double the investment with continuous compounding, we use the formula:
A = Pe^{rt}
Since A = 2P, we solve for t:
t = \frac{1}{r} \cdot \log(2)
Substituting r = 0.02, we find the time. After performing the calculations and rounding to two decimal places, we find that with quarterly compounding, the investment doubles in about 35 years, and with continuous compounding, it doubles in about 34.66 years.