Respuesta :
The population growth in this question is assumed to be exponential.
The exponential formula is given to be:
[tex]y=a(b)^x[/tex]where a is the initial amount and b is the growth rate of the population.
The question provides the following parameters:
[tex]a=100[/tex]We can find the growth rate by making the following substitution: If there were 230 insects after 7 days, we have:
[tex]x=7,y=230[/tex]Therefore,
[tex]\begin{gathered} 230=100(b)^7 \\ b^7=2.3 \\ b=\sqrt[7]{2.3} \\ b=1.126 \end{gathered}[/tex]Therefore, the exponential model for the problem is:
[tex]y=100(1.126)^x[/tex]To find the time it takes for the population to reach 600 insects, we can substitute for y = 600 into the model and solve for x:
[tex]\begin{gathered} 600=100(1.126)^x \\ 1.126^x=\frac{600}{100} \\ 1.126^x=6 \end{gathered}[/tex]Finding the natural logarithm of both sides:
[tex]\ln 1.126^x=\ln 6[/tex]Recall the law of logarithms:
[tex]\ln x^a=a\ln x^{}[/tex]Therefore,
[tex]\begin{gathered} x\ln 1.126=\ln 6 \\ \therefore \\ x=\frac{\ln 6}{\ln 1.126} \\ x=15.098\approx15 \end{gathered}[/tex]Therefore, it will take 15 days for the insect population to reach 600 insects.
