The rate of growth of a population of micro-organisms is modelled by the equation: dP/dt = 3t^2+6t, where P is the population size at time t hours. Given that P=100 at t=1, find P in terms of t.

First, we integrate the equation with respect to t to find an equation for P. dP/dt = 3t² + 6t Then, P= integral (3t² + 6t) dt Integrating gives P= t³+3t²+c, c is the constant of integration. As we are given the boundary condition P=100 when t=1, sub in these values into the equation for P to find what c is. 100=1³+3(1²) +c Gives c=96 We get an equation for P with the correct value of c, P=t³+3t²+96