Respuesta :
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
[tex]P(t)=0.3t^2+6t+80[/tex]
First we find out the initial population, i.e. t=0,
So, our quadratic equation becomes,
[tex]P(0)=0+0+80=80[/tex]
According to question, we have said that if the population has twice its initial population, then it becomes
[tex]P(t)=2\times 80=160[/tex]
So, time taken to reach this above mentioned population is given by
[tex]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]
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.
