Question

14. A bacteria culture is started with 350 bacteria. After 4 hours, the population has grown to 732 bacteria. If the population grows exponentially according to the formula Pt=P0(1+r)t (a) Find the growth rate. Round your answer to the nearest tenth of a percent.r = %(b) If this trend continues, how many bacteria will there be in one day? bacteria(c) How long will it take for this culture to triple in size? Round your answer to the nearest tenth of an hour. hours

Answer 1

$p_t=p_{\circ}(1+r)^t$

a.

$\begin{gathered} p_{\circ}=350 \\ p_t=732 \\ t=4\text{ hours} \\ r=? \\ \frac{p_t}{p_{\circ}}=(1+r)^t \\ \frac{732}{350}=(1+r)^4 \\ 2.09142857143=(1+r)^4 \\ \log 2.09142857143=4\log (1+r) \\ \frac{0.32035403281}{4}=\log (1+r) \\ 0.0800885082=\log (1+r) \\ 10^{0.0800885082}=1+r \\ 1.20250945556-1=r \\ r\approx0.20250945556\approx20.3\text{ \%} \end{gathered}$

b.

$\begin{gathered} p_t=350(1+0.203)^{24} \\ p_t=350(1.203)^{24} \\ p_t=350\times84.4063375148=29542.2181302\approx29542\text{ bacteria} \end{gathered}$

c.

$\begin{gathered} 350\times3=350(1+0.203)^t \\ \frac{1050}{350}=1.203^t \\ 3=1.203^t \\ \log 3=t\log 1.203 \\ t=\frac{0.47712125472}{0.08026562733} \\ t=5.94427864744 \\ t\approx5.9\text{ hours} \end{gathered}$