Respuesta :
Formula:
[tex]\begin{gathered} P_t=P_o(1+r)^t \\ \end{gathered}[/tex]a) Given:
[tex]\begin{gathered} P_t=675 \\ P_o=500 \\ t=4 \\ r=\text{?} \end{gathered}[/tex]Substitute the given into the formula to find the rate:
[tex]\begin{gathered} 675=500(1+r)^4_{} \\ \frac{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}[/tex]b) Substitute the given below into the formula:
[tex]\begin{gathered} r=7.8\text{ \%= 0.078} \\ P_t=\text{?} \\ P_o=500 \\ t=24\text{ hours (1 day)} \end{gathered}[/tex][tex]\begin{gathered} P_t=500(1+0.078)^{24} \\ P_t=500\times6.06527 \\ P_t=3032.635\text{ } \end{gathered}[/tex]There will be approximately 3033 bacteria in one day.
c) Given:
[tex]\begin{gathered} P_o=500 \\ P_t=1500 \\ t=\text{?} \\ r=7.8\text{ \% = 0.078} \end{gathered}[/tex]We will calculate for how long it will take the culture to triple thus:
[tex]\begin{gathered} 1500=500(1+0.078)^t \\ 3=(1.078)^t \\ \ln 3=t\ln (1.078) \\ \frac{\ln3}{\ln1.078}=t \\ 14.627=t \\ 14.6\text{hours (to the nearest tenth of an hour) = t} \end{gathered}[/tex]