A bacteria culture initially contains 300 cells

Question

A bacteria culture initially contains 300 cells and grows at a rate proportional to its size. After 4 hours the population has increased to 320. (a) Find an expression for the number of bacteria after  hours.   (b) Find the number of bacteria after 6 hours. (c) Find the rate of growth after 6 hours. (d) When will the population reach 8000?

Answer 1

Let h = the number of hours and NB = the number of bacteria. Then we have

NB = 300 x ( 305/300 )^h So that

NB(6) = 300 x ( 305/300 )⁶ = 331 So the growth rate is 331/300 = 1.103 or 11.03%

To find out how many hours for the population to reach 8000 we have:

300 x ( 305/300)^h = 8000 Take the natural log of both sides of the equation:

ln( 305/300 )^h = ln(8000) Using the properties of ln we have

ln(300) + h x ln( 305/300) = ln(8000) So we have

5.70378 + .016529 x h = 8.98720 or .016529 x h = 3.28341 Finally dividing

h = 198.64 hours

Answer 2