Question

The number of bacteria in a culture is modeled by N(t) = 1500e^0.31t Where t is given in hours. A) The hourly rate of growth of the bacterium population is _______B) After how many hours will the number of bacteria reach 10,000? Your answer is __________

Answer 1

The bacterial will reach 10, 000 after 6.11 hours

Explanation:

Part B

`\begin{gathered} Given\text{ that} \\ N(t)\text{ = 1500 }e^(0.31t) \\ \text{After how many hours will the number of bacteria reach 10, 000} \\ \text{let N(t) = 10, 000} \\ t\text{ = time in hours} \\ 10,000\text{ = 1500}e^(0.31t) \\ \text{Divide both sides by 1500} \\ (10,000)/(1,500)\text{ = }(1500)/(1500)e^(0.31t) \\ 6.666\text{ = }e^(0.31t) \\ \text{Take the natural logarithms of both sides} \\ \ln (6.666)\text{ = }\ln e^(0.31t) \\ \ln (6.666)\text{ = 0.31t} \\ 1.897\text{ = 0.31t} \\ \text{Divide both sides by 0.31} \\ (1.897)/(0.31)\text{ = }(0.31t)/(0.31) \\ t\text{ = 6.11 hours} \\ \text{Hence, the bacteria will reach 10, 000 after 6.11 hours} \end{gathered}`