A model for the population $P(t)$ in a suburb of a large city is given by the initial$-$value problem

$\frac{dP}{dt}=P({10}^{-1}-{10}^{-7}P),P(0)=6,000$

where $t$ is measured in months.

Find the population $P$ of the suburb at time $t.$

$P(t)=\frac{3000000e^{(\frac{t}{10})}}{497\frac{t}{10}}$

What is the limiting value of the population?

$$ Great!

At what time $($in months$)$ will the population be equal to one$-$half of this limiting value? $($Round your answer to one decimal place.$)$

$$ months