Question

12. A bacteria culture is started with 500 bacteria. After 4 hours, the population has grown to 675 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

Formula:

`\begin{gathered} P_t=P_o(1+r)^t \\ \end{gathered}`

a) Given:

`\begin{gathered} P_t=675 \\ P_o=500 \\ t=4 \\ r=\text{?} \end{gathered}`

Substitute the given into the formula to find the rate:

`\begin{gathered} 675=500(1+r)^4_{} \\ (675)/(500)=(1+r)^4 \\ 1.35=(1+r)^4 \\ \sqrt[4]{1.35}=\sqrt[4]{(1+r)^4} \\ 1.077912=(1+r) \\ 1.077912-1=r \\ 0.077912=r \\ 7.77912\text{ \%=r} \\ 7.8\text{ \%=r (to the nearest tenth)} \end{gathered}`

b) Substitute the given below into the formula:

`\begin{gathered} r=7.8\text{ \%= 0.078} \\ P_t=\text{?} \\ P_o=500 \\ t=24\text{ hours (1 day)} \end{gathered}`
`\begin{gathered} P_t=500(1+0.078)^(24) \\ P_t=500*6.06527 \\ P_t=3032.635\text{ } \end{gathered}`

There will be approximately 3033 bacteria in one day.

c) Given:

`\begin{gathered} P_o=500 \\ P_t=1500 \\ t=\text{?} \\ r=7.8\text{ \% = 0.078} \end{gathered}`

We will calculate for how long it will take the culture to triple thus:

`\begin{gathered} 1500=500(1+0.078)^t \\ 3=(1.078)^t \\ \ln 3=t\ln (1.078) \\ (\ln3)/(\ln1.078)=t \\ 14.627=t \\ 14.6\text{hours (to the nearest tenth of an hour) = t} \end{gathered}`