To solve this question we are going to need to define an equation with the data that the problem gives us
The first part (rate of growth) shows us a derivate, this is equal to 9 square root of t or:
[tex]\frac{dP(t)}{dt}=9\sqrt{t}[/tex]
where P is the population size and t is the time in days
Now we need the population after 7 days, so we need the P(t) formula, which means that we need the integration
First, we send the dt to the other side to have something to integrate
[tex]\int dP(t)=\int9\sqrt{t}dt[/tex]
Solving
[tex]\begin{gathered} P(t)=9(\frac{2}{3}t^{\frac{3}{2}})+C \\ \\ P(t)=6t^{\frac{3}{2}}+C \end{gathered}[/tex]
We know that P(0)=600 so
[tex]P(0)=600=C[/tex][tex]P(t)=6t^{\frac{3}{2}}+600[/tex]
Now we replace the t by 7
[tex]\begin{gathered} P(7)=6(7^{\frac{3}{2})}+600 \\ P(7)=711.121 \end{gathered}[/tex]
Answer: 711
