If P'(x) is the rate of growth of the population, the actual population is given by the function P(x); therefore, we need to integrate P'(x) as shown below
[tex]P^{\prime}(t)=\sqrt{t}(4170t+5430)=t^{\frac{1}{2}}(4170t+5430)=4170t^{\frac{3}{2}}+5430t^{\frac{1}{2}}[/tex]
Thus,
[tex]\begin{gathered} \Rightarrow P(t)=\int P^{\prime}(t)dt=\int(4170t^{\frac{3}{2}}+5430t^{\frac{1}{2}})dt=4170\int t^{\frac{3}{2}}dt+5430\int t^{\frac{1}{2}}dt \\ =4170(\frac{2}{5}t^{\frac{5}{2}})+5430(\frac{2}{3}t^{\frac{3}{2}})+C=1668t^{\frac{5}{2}}+3620t^{\frac{3}{2}}+C \\ C\rightarrow constant \end{gathered}[/tex]
Therefore, calculating the population increase from t=1 to t=4,
[tex]\begin{gathered} \Rightarrow P(4)-P(1)=1668((4)^{\frac{5}{2}}-(1)^{\frac{5}{2}})+3620((4)^{\frac{3}{2}}-(1)^{\frac{3}{2}})+C-C=1668(31)+3620(7) \\ =77048 \end{gathered}[/tex]
Thus, the answer is 77048
