Jamie invests $500 in an account that is compound continuously at an annual rate of 5% according to the formula A=Pe^rt, where A is the amount accrued, P is the principal, r is the rate of interest and t is the time in years. approximately how many years will it take Jamie's money to double?
1000=500e^(.05)t
2=e^.05t
ln2=.05t
t=13.9
yes
To solve this problem, we can use the formula for continuous compound interest:
A = Pe^(rt)
In this case, Jamie invests $500, so P = $500. The annual interest rate is 5% or 0.05, so r = 0.05. We need to find the value of t, the time it takes for the investment to double.
We can rewrite the formula as:
2P = Pe^(rt)
Dividing both sides of the equation by P, we get:
2 = e^(rt)
Now, we can take the natural logarithm (ln) of both sides to isolate the exponent term:
ln(2) = rt ln(e)
Since ln(e) is equal to 1, the equation simplifies to:
ln(2) = rt
Now, we can solve for t. Divide both sides of the equation by r:
t = ln(2) / r
Substituting the values, we get:
t = ln(2) / 0.05
Using a calculator, we find that ln(2) ≈ 0.693147181, so:
t ≈ 0.693147181 / 0.05 ≈ 13.9
Therefore, it will take approximately 13.9 years for Jamie's money to double.