The Australian Lord Howe Island stick insects may be the rarest insects not already extinct in the world. It is believed there may be only a few of them (less than 30) left in the wild in th world. In a last-ditch effort to expand the population safely, these stick insects are carefully bred and introduced back into their native habitat on a very careful linear schedule. Assume that a few weeks ago, 40 of the stick insects (P0 ) were introduced on Lord Howe Island by scientists into the existing habitat. If natural processes occur correctly, each week, the population grows following the linear growth model. 4 weeks later, there were 75 of the stick insects (P4 ) on the island. a) Find an explicit formula for the stick insect population Py after N weeks have passed. b) Following your formula, how long will it take the stick insect population to grow to 350. Give the answer accurate to 3 decimal places. c) If the population continues to grow following this linear model, what is the rate of growth? Explain in words what this means.

[tex]\begin{gathered} let\text{ y be the number of stick insects and x the w}eeks,then\text{ } \\ y=mx+b \\ is\text{ the linear behavior.} \\ In\text{ this case when y=40 insects we start counting, hence, x=0,Therefore,} \\ 40=m(0)+b \\ \text{hence, b=40. Then the linear equation has the form} \\ y=mx+40 \\ Now.\text{ me must find the slope m. In} \end{gathered}[/tex][tex]\begin{gathered} In\text{ order to find the slope, we can use the given data:} \\ \text{when x=4 w}eeks\text{ then y=75 insects. By substituying these data into the linear equations, one has} \\ 75=m(4)+40 \\ 75=4m+40 \\ 4m=75-40 \\ 4m=35 \\ m=\frac{35}{4} \\ m=8.75 \\ \text{Hence, the linear behavior is given by} \\ y=8.75x\text{ + 40} \end{gathered}[/tex][tex]\begin{gathered} b)\text{ In this case y=350 insects and we must find x. that is} \\ 350=8.75x+40 \\ 350-40=8.75x \\ 8.75x=310 \\ x=\frac{310}{8.75} \\ x=35.429\text{ w}eeks \end{gathered}[/tex][tex]\begin{gathered} c)\text{ the rate of growing is given by the slope. Then, the rate is} \\ m=8.75=\frac{35}{4} \end{gathered}[/tex]