a) The initial population of dandelion can be found by finding p(0), i.e. by substituting t=0 into p. The initial population is
b) The rate at which the population changes after 5 years is found by substituting t=5 into the expression for . The simplest way to find is to write p as
by dividing top and bottom by . We can then use the chain rule to differentiate p:
See more on differentiating e to the x. Substituting t=5 into this expression gives
c) From part b), we have that p can be written as
Recall that is a number between 0 and 1 for any positive x value. It follows that is between 0 and 1 (not including) for any positive t value. Hence, is a number between 4 and 5 and so we are dividing 500 by a number that is between 4 and 5. The result is a number between 100 and 125 and so the population does not exceed 125.