Let us call the size of the colony y(t), where y is a function of the time t. Therefore the size of the colony depends on the time. Then call the rate of growth y'(t), were t is the time again. We're given that the rate of growth is proportional to the size of the colony, call this proportional factor k. Hence we have
y'(t) = k*y(t) <=> y'(t) - k*y(t) = 0Solving this differential equation gives us the general solution
y(t) = C*e^(kt)
Use the general solution to solve the value of k using the fact the the colony triples every third hour. We also know that when t = 0 we have
y(0) = C*e^(k0) = CThis means that C is the initial value of the problem which in this case is the population of the bacteria colony.