Question

A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 270. (a) Find an expression for the number of bacteria after t hours. P(t) = (b) Find the number of bacteria after 4 hours. (Round your answer to the nearest whole number.) P(4) = bacteria (c) Find the rate of growth after 4 hours. (Round your answer to the nearest whole number.) P'(4) = bacteria per hour (d) When will the population reach 10,000? (Round your answer to one decimal place.) t = hr

Answer 1

Answer:
`\bold{a)\ P(t)=P_o\cdot e^(t\cdot ln(2.7))}`

b) 5314

c) ln 2.7

d) 4.6 hrs

Explanation:

`P(t) = P_o\cdot e^(kt)\\\\\bullet \text{P(t) is the number of bacteria after t hours} \\\bullet P_o\text{ is the initial number of bacteria}\\\bullet \text{k is the rate of growth}\\\bullet \text{t is the time (in hours)}\\\\\\270=100\cdot e^(k(1))\\2.7=e^k\\ln\ 2.7=\ln e^k\\\boxed{ln\ 2.7=k}\\\\\text{So the equation to find the number of bacteria is: }\boxed{P(t)=P_o\cdot e^(t\cdot ln(2.7))}\\\\\\P(4)=100\cdot e^(4\cdot ln(2.7))\\.\qquad =\boxed{5314}`

`10,000=100\cdot e^(t\cdot ln(2.7))\\100=e^(t\cdot ln(2.7))\\ln\ 100=ln\ e^(t\cdot ln(2.7))\\ln\ 100=t\cdot ln(2.7)\\(ln\ 100)/(ln\ 2.7)=t\\\\\boxed{4.6=t}`