A bacteria culture initially contains 100 cells and grows at a rate proportional

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 490. (a) Find an expression for the number of bacteria after t hours. (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.) hr Need Help? achTalk to a Tetor

Explanation / Answer

If the rate of growth of the population is proportional to the population, we know that the growth will be exponential, so the number of cells after t hours will have the form P(t)= A ebt. In this formula, we know that A=P(0) , the initial population, so A=100. To find b, we plug in our other data point: P(1)=490=100 eb.1 => eb=4.9 => b=ln(4.9)

b= ln(4.9) = 1.5892

1. The number of cells after t hours is P(t)= 100 ebt= 100 ebt where b is calculated above.

2.The number of bacteria after 4 hours

P(4)= 100 eb.4 =57,640

3. To calculate dP/dt at 4 hours, dP/dt =bAebt= bP

dP/dt(4 hours)=ln4.9 . 100e4 ln(4.9)= (4.9)4. 100. ln(4.9)

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