In order to solve this problem, we will use the formula for continuous compounding, which is A = Pe^(rt), where:
- A is the final amount,
- P is the principal amount (the initial amount of money),
- r is the annual interest rate, and
- t is the time in years.
We are given an initial investment P, an interest rate r of 7.75%, and we want to find out the time it takes for the investment to double. So, we want A to be twice the initial amount, that is, A = 2P.
Substituting A = 2P into the initial formula gives us the equation 2P = Pe^(rt). We can simplify this equation by dividing both sides by P, which gives us:
2 = e^(rt)
Next, we need to solve this equation for t. To do this, we'll use a logarithm to eliminate the exponential function e^x.
By applying the natural logarithm ln to both sides of the equation, we obtain:
ln(2) = rt.
And finally, rearranging to solve for t gives us:
t = ln(2) / r.
Before substituting our values, we have to convert our annual interest rate from percentage to a decimal. The rate r is thus 7.75 / 100 = 0.0775.
Substituting r into our equation yields:
t = ln(2) / 0.0775.
Evaluating this expression, we find
t ≈ 8.944.
This is the number of years it will take for the investment to double under continuous compounding at an annual interest rate of 7.75%.