The formula to find the amount of an investment earning interest at the rate of r, compounded continuously, after t years is given by:
A = P * e^(r*t)
Where A is the amount after t years, P is the initial amount, r is the rate of interest, and e is the mathematical constant approximately equal to 2.71828.
We are given that the investment will double in t years. This means that the amount after t years will be twice the initial amount, or:
2P = P * e^(r*t)
Simplifying, we can divide both sides by P:
2 = e^(r*t)
To solve for t, we can take the natural logarithm of both sides:
ln(2) = ln(e^(r*t))
Using the logarithmic identity ln(e^x) = x, we get:
ln(2) = r*t*ln(e)
Since ln(e) = 1, we can simplify further:
ln(2) = r*t
Finally, we can solve for t by dividing both sides by r and rounding the answer to the nearest whole number:
t = ln(2) / r ≈ 6.93 / 0.1 ≈ 69.3 ≈ 69 years
Therefore, an investment earning interest at the rate of 10%, compounded continuously, will double in approximately 69 years.