A farm purchased in 2000 for $\$ 1$ million was valued at $\$ 5$ million in 2012. If the farm continues to appreciate at the same rate (with continuous compounding), when will it be worth $\$ 10$ million?
Find a function $A(t)$ that gives the current value of the farm.
$A(t)=\square$ (Type an exact answer.)
The farm will be worth $\$ 10$ million dollars in the year
(Do not round until the final answer. Then round to the nearest year as needed.)

Step 1 ：We are given that the farm was purchased in 2000 for $1 million and was valued at $5 million in 2012. This is a problem of continuous compound interest, which can be modeled by the formula: \(A(t) = P * e^{rt}\), where \(A(t)\) is the amount of money accumulated after n years, including interest, \(P\) is the principal amount (the initial amount of money), \(r\) is the annual interest rate (in decimal), and \(t\) is the time the money is invested for, in years.

Step 2 ：We can use the given values to solve for the rate of appreciation (r). We know that the initial value of the farm (P) is $1 million, and after 12 years (t = 12), the value of the farm (\(A(t)\)) is $5 million.

Step 3 ：Once we have the rate of appreciation, we can use it to find the time (t) when the farm will be worth $10 million.

Step 4 ：The calculated time when the farm will be worth $10 million is approximately 17.17 years from the year 2000. This means the farm will reach this value in the year 2000 + 17.17 = 2017.17. Since we need to round to the nearest year, this will be the year 2017.

Step 5 ：Final Answer: The farm will be worth $10 million dollars in the year \(\boxed{2017}\).