There are 400 rice weevils at the beginning of an insect study. The population is expected to grow at a rate of 150% each week.a.) Write and simplify a formula that can be used to predict the rice weevil population for any week after the beginning of the study.b.) Use the formula to predict the rice weevil population 10 weeks after the beginning of the study. Show your work.c.) Use the formula to predict when the rice weevil population will reach 1,000,000,000. Show your work.

Answer:Part a) [tex]y=400(2.5)^{x}[/tex]Part b) [tex]3,814,697\ rice\ weevil\ population[/tex]Part c)  [tex]16.1\ weeks[/tex]Step-by-step explanation:Part a) Write and simplify a formula that can be used to predict the rice weevil population for any week after the beginning of the studywe know that100%+150%=250%=250/100=2.5In this problem we have an exponential function of the form[tex]y=a(b)^{x}[/tex]wherea is the initial valueb is the basey ----> is the rice weevil  populationx ----> the number of weeksIn this problema=400 rice weevilsb=2.5substitute[tex]y=400(2.5)^{x}[/tex]Part b) Use the formula to predict the rice weevil population 10 weeks after the beginning of the study. Show your workwe have [tex]y=400(2.5)^{x}[/tex]soFor x=10 weekssubstitute[tex]y=400(2.5)^{10}=3,814,697\ rice\ weevil\ population[/tex]Part c) Use the formula to predict when the rice weevil population will reach 1,000,000,000.we have [tex]y=400(2.5)^{x}[/tex]soFor y=1,000,000,000 rice weevil  populationsubstitute in the formula and solve for x [tex]1,000,000,000=400(2.5)^{x}[/tex]Simplify [tex]2,500,000=(2.5)^{x}[/tex]Apply log both sides [tex]log(2,500,000)=x*log(2.5)[/tex] [tex]x=log(2,500,000)/log(2.5)=16.1\ weeks[/tex]