how long will it take $40,000 invested at 9% compounded continuously to double A=Pe^rt
2 = e^.09 t
ln 2 = .09 t = .6931
t = 7.7 years
To find out how long it will take for $40,000 invested at 9% compounded continuously to double, you can use the formula A = Pe^rt, where:
A is the final amount
P is the principal amount (initial investment)
e is Euler's number (approximately 2.71828)
r is the interest rate (as a decimal)
t is the time (in years)
In this case, the final amount is double the initial investment, so A = 2P. And the interest rate is 9%, which should be converted to a decimal by dividing it by 100 (9/100 = 0.09).
The formula becomes:
2P = Pe^(0.09t)
Next, we can simplify the equation by canceling out the principal amount P on both sides:
2 = e^(0.09t)
Now, we need to isolate the variable t to solve for it. We take the natural logarithm (ln) of both sides of the equation. The natural logarithm is the inverse function of e.
ln(2) = ln(e^(0.09t))
Using the property of logarithms that ln(e^x) = x, the equation becomes:
ln(2) = 0.09t
Finally, divide both sides of the equation by 0.09:
t = ln(2) / 0.09
Using a calculator, find the natural logarithm of 2 (ln(2)), then divide the result by 0.09. The value you get is the time it will take for the investment to double.
Please note that the answer will be in years.