(1 point) If 6000 dollars is invested in a bank account at an interest rate of 9 per cent per year, compounded continuously. How many years will it take for your balance to reach 40000 dollars? NOTE: Give your answer to the nearest tenth of a year.
Solution
Step 1 ：Given that the principal amount (P) is $6000, the final amount (A) is $40000, and the interest rate (r) is 9% per year or 0.09 in decimal form.
Step 2 ：We are asked to find the time (t) it will take for the balance to reach $40000.
Step 3 ：We can use the formula for continuous compound interest, which is \(A = P * e^{rt}\), where A is the final amount, P is the principal amount, r is the interest rate, and t is the time in years.
Step 4 ：We can rearrange this formula to solve for t: \(t = \frac{ln(A/P)}{r}\).
Step 5 ：Substituting the given values into the formula, we get \(t = \frac{ln(40000/6000)}{0.09}\).
Step 6 ：Solving this equation gives us \(t \approx 21.1\).
Step 7 ：Final Answer: It will take approximately \(\boxed{21.1}\) years for the balance to reach $40000.
