answer: at 24 february, 2014 the population gets twice of its initial population.
explanation:
since we have given that
the population for alpha city, t years after january 1, 2004 is given as
![p(t)=0.3t^2+6t+80](/tex.php?f=p(t)=0.3t^2+6t+80)
first we find out the initial population, i.e. t=0,
so, our quadratic equation becomes,
![p(0)=0+0+80=80](/tex.php?f=p(0)=0+0+80=80)
according to question, we have said that if the population has twice its initial population, then it becomes
![p(t)=2\times 80=160](/tex.php?f=p(t)=2\times 80=160)
so, time taken to reach this above mentioned population is given by
![160=0.3t^2+6t+80\\\\160-80=0.3t^2+6t\\\\80=0.3t^2+6t\\\\0.3t^2+6t-80=0\\\\\text{using quadratic formula ,we get }\\\\t_{1,\: 2}=\frac{-60\pm \sqrt{60^2-4\cdot \: 3\cdot \: 800}}{2\cdot \: 3}\\\\t=-29.14\ and\ t=9.149](/tex.php?f=160=0.3t^2+6t+80\\\\160-80=0.3t^2+6t\\\\80=0.3t^2+6t\\\\0.3t^2+6t-80=0\\\\\text{using quadratic formula ,we get }\\\\t_{1,\: 2}=\frac{-60\pm \sqrt{60^2-4\cdot \: 3\cdot \: 800}}{2\cdot \: 3}\\\\t=-29.14\ and\ t=9.149)
now, we know that time can't be negative so, we take t=9.149 years.
hence, at 24 february, 2014 the population gets twice of its initial population.