Dwight is saving up money for a down payment on a house. He currently has $\$ 3888$, but knows he can get a loan at a lower interest rate if he can put down $\$ 4849$. If he invests the $\$ 3888$ in an account that earns $4.7 \%$ annually, compounded continuously, how long will it take Dwight to accumulate the $\$ 4849$ ? Round your answer to two decimal places, if necessary. Answer How to enter your answer (opens in new window) years

Step 1 ：The problem is asking for the time it will take for an initial amount of money to grow to a certain amount under continuous compounding. The formula for continuous compounding is \(A = Pe^{rt}\), where \(A\) is the final amount, \(P\) is the principal amount, \(r\) is the interest rate, and \(t\) is the time in years.

Step 2 ：We can rearrange this formula to solve for \(t\): \(t = \ln(A/P) / r\).

Step 3 ：In this case, \(A = 4849\), \(P = 3888\), and \(r = 0.047\). We can plug these values into the formula to find \(t\).

Step 4 ：Substituting the given values into the formula, we get \(t = \ln(4849/3888) / 0.047\).

Step 5 ：Solving the above expression, we get \(t = 4.6995236494685635\).

Step 6 ：Rounding to two decimal places, we get \(t = 4.70\).

Step 7 ：Final Answer: The time it will take Dwight to accumulate \$ 4849 is approximately \(\boxed{4.70}\) years.