The formula for continuous compound interest is A = P * e^(rt), 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 interest rate (in decimal).
- t is the time the money is invested for (in years).
In this case, we want to find the time when the amount becomes double the principal. So, our equation will be 2 * P = P * e^(rt). After simplifying, we get e^(rt) = 2.
Taking the natural logarithm (ln) of both sides to eliminate the exponential, we get:
ln(e^(rt)) = ln(2).
The left-hand side simplifies to r*t (since ln and e are inverse of each other), leading to:
r*t = ln(2).
We can solve for t (time) now by dividing both sides by r:
t = ln(2) / r.
To convert the rate of interest given in the question from the percentage, we divide it by 100. So r (in decimal form) is 2.1 / 100 = 0.021.
Substitute r into the equation:
t = ln(2) / 0.021.
Calculate this expression to find the time it takes for the amount to double. After calculations, you should obtain a result:
t ≈ 33.01.
This means it will take approximately 33.01 years for the initial investment to double under the given conditions.