Answer
(a) -- 32000 bacteria
(b) -- $f(t)=500(2)^{2t}$
(c) -- 1260 bacteria
(d) -- $t=3.82$ hours
Work Step by Step
We are told that a culture of bacteria starts with 500 bacteria present, and this number doubles every half-hour. Thus we have: $$f(t)=500(2)^{2t}$$ (a) We are asked to find the number of bacteria after 3 hours. We plug in $t=3$:$$f(6)=500(2)^{2*3}=32000$$ (b) We are asked to find the number of bacteria after $t$ hours. This is essentially asking us for the equation that we have already created. $$f(t)=500(2)^{2t}$$ (c) We are asked to find how many bacteria there are after 40 minutes have elapsed. Because $40/60=\frac{2}{3}$, we must plug in a value of $\frac{2}{3}$ for t. $$f(\frac{2}{3})=500(2)^{2*\frac{2}{3}}=1259.921\approx 1260$$ (d) In this last part, we are asked to graph the population function (found in part (b)) and estimate the time for the population to reach $100,000$. If we create a second equation that represents this population value $g(t)=100,000$, we can determine the exact point of intersection. Using a graphing utility, we can find that the two functions intersect at $t=3.82$ hours.
