A population p ( t ) of 1000 bacteria introduced into a nutrient medium grows according to the relation p ( t )=1000+ 1000 t 100+ t ^2 . The maximum size of this bacterial population is

Step-by-step solution

Maximum bacterial population To determine the maximum population size, maximize the function p(t) = 1000 + 1000t 100+t^2 . The maximum of p(t) occurs at the same t as the maximum of 1000t 100+t^2 , since 1000 is constant. Differentiate this function: f'(t) = 1000(100+t^2) - 1000t(2t) (100+t^2)^2 = 100000 - 1000t^2 (100+t^2)^2 Set the derivative to zero to find critical points. Only t = 10 is physically meaningful in the domain t 0 . The population at t = 10 is p(10) = 1000 + 1000 10 100+100 = 1000 + 50 = 1050 . Sin