(please show how you got answer)(please read the more information box)

Question

the rate of growth of a particular population is given by (dp/dt) = 50t^2 - 100t^3/2 ,where (P) is the population size and (t) is the time in years. The initial utility to graph the function, and then use the graph to estimate how many years it will take for the population to reach 50,000.

Answer 1

dp/dt=50t^2-100t^3/2, so dp=(50t^2-100t^3/2)dt. Integrating we get p=(50t^3)/3-(200/5)t^5/2+p0, where p0 is an initial population. We have no indication of p0, so we have to assume it's zero or negligible at t=0. So p=(50t^3)/3-40t^(5/2)=50000. Using a calculator, t=18.856 years for the population to reach 50000.

Using a graph it is clear that at t=0, p=0, and between p=0 and 5.76 the population is negative. After t=5.76 years the population rises steadily until at t=18.86 it reaches 50000. At t=4 the graph appears to have a minimum at p=-213.33. To offset this p0 could be set to 213.33 so that the minimum is zero at t=4 years. When this adjustment is made the population reaches 50000 at t=18.836 years approximately.