Final answer:
It takes approximately 17.33 years for $5,900 to double at an annual interest rate of 4%, compounded continuously. We use the continuous compounding formula A = Pert, solve for t, and get t ≈ 17.33 when A is twice P.
Step-by-step explanation:
To find the time it takes for $5,900 to double when invested at an annual interest rate of 4%, compounded continuously, we can use the formula for continuous compounding:
A = Pert
where:
A is the final amount that the initial investment grows to after interest.
To double the money, A will be 2 times P ($5,900). So, our equation becomes:
2 * $5,900 = $5,900 * e0.04t
Solving for t, we divide both sides by $5,900, getting:
2 = e0.04t
Take the natural logarithm of both sides:
ln(2) = 0.04t
Divide by 0.04 to isolate t:
t = ln(2) / 0.04
Finally, calculate t:
t ≈ 17.33 years
So, it takes approximately 17.33 years for $5,900 to double at an annual interest rate of 4%, compounded continuously.