The population of a small town is modeled by the equation $P=1450 e^{6.5 t}$ where $t$ is measured in years. In approximately how many years will the town's population reach 20,000 ? (Round your answer to two decimal places.)
$\mathrm{yr}$

Step 1 ：We are given the population model \(P=1450 e^{6.5 t}\) and we are asked to find the time \(t\) when the population \(P\) will be 20,000.

Step 2 ：We can solve this by setting \(P=20000\) and solving for \(t\).

Step 3 ：This will involve taking the natural logarithm of both sides of the equation to isolate \(t\).

Step 4 ：Substituting the given values into the equation, we get \(20000 = 1450 e^{6.5 t}\).

Step 5 ：Solving for \(t\), we get \(t = 0.4037182641725397\).

Step 6 ：Rounding to two decimal places, we get \(t = 0.40\).

Step 7 ：Final Answer: The town's population will reach 20,000 in approximately \(\boxed{0.40}\) years.