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.