Final answer:
To determine the time it takes for $6,000 to double at a 3% continuously compounded interest rate, we use the continuous compounding formula. Solving the formula, we find that it takes approximately 23.1 years for the money to double.
Step-by-step explanation:
The question asks how long it will take for $6,000 to double in a bank account with compound interest at a rate of 3% compounded continuously. To solve this type of problem, we use the formula for continuous compounding, which is A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (as a decimal), and t is the time (in years).
In this scenario, you want to double your money, which means A should be $12,000 (double of $6,000), P is $6,000, r is 0.03 (3% as a decimal), and we are solving for t. The equation becomes $12,000 = $6,000e0.03t. To find t, we divide both sides by $6,000, which simplifies the equation to 2 = e0.03t. Then, we take the natural logarithm (ln) of both sides which simplifies to ln(2) = 0.03t. Finally, divide ln(2) by 0.03 to solve for t.
t = ln(2) / 0.03 ≈ 23.1 years
So, it will take approximately 23.1 years for the money to double at a 3% continuously compounded interest rate, which corresponds to option A).