Question

in the following, write an algebraic expression in terms of the given variables that represent the indicated quantity....The amount of bacteria after N min, if the initial amount of bacteria is Q and the amount of bacteria triples every 20 sec. (Hint: the answer should contain q as well as n).

Answer 1

Let q be the amount of bacteria after n minutes. We know that the amount of bateria triples every 20 seconds, this means that in one minute the amount of bacteria will triples 3 times, that is the amount triples every:

`\frac{1}{3\text{ }}\text{ min}`

The expression we are looking for has to be of the form:

`q(n)=Q(3^(kn))`

So we need to find the coefficient k. To do this we know that after 1/3 of a minute the amount will be 3Q, then we have:

`\begin{gathered} q(1)=Q(3^{(1)/(3)k}) \\ 3Q=Q(3^{(1)/(3)k}) \\ 3=3^{(1)/(3)k} \\ \text{which implies that:} \\ (1)/(3)k=1 \\ k=3 \end{gathered}`

Therefore the equation we are lokking for is:

`q(n)=Q^{}(3^(3n))`