Question

The initial size of a culture of bacteria is 1500. After one hour the bacteria count is 6000.

A.) Find a function n(t) = n₀e^rt that models the population after t hours (round your r value to five decimal places.) n(t)=

B.) Find the population after 1.5 hours. (round answer to nearest whole number.) n(1.5)=

C.) After how many hours will the number of bacteria reach 10,000? (round your answer to one decimal place.)

Answer 1

A.) Let the function modelling the growth of the bacteria be n(t) = n₀e^rt , where n₀ is the initial size of the culture of bacteria, n(t) its size after t hours and r is the constant of growth. Here, n₀ is 1500, and when t = 1, we have n(1) = 1600 so that 1600 = 1500e^1*r or, e^r = 1600/1500 = 16/15. Now, on taking natural log of both the sides, we get ln (e^r) = ln(16/15) or, r (ln e )= ln 16 -ln 15 or, r = 2.772588722 -2.708050201 = 0.064538521 or, 0.06454 (on rounding off to five decimal places). Thus, n(t) = 1500e^0.06454t.

B.) When t = 1.5 hour, the population of bacteria after 1.5 hours is 1500 e^0.06454*1.5 = 1500 e^0.09681 = 1500*1.10165104 = 1652 (on rounding off to the nearest whole number).

C.) Let the number of bacteria reach 10,000 after t hours. Then 10000 = 1500e^0.06454t or, e^0.06454t = 10000/1500 = 20/3. On taking natural log of both the sides, we get 0.06454t = ln 20-ln 3 = 2.995732274-1.098612289 = 1.897119985. Hence t = 1.897119985/0.06454 = 29.4 hours(on rounding off to one decimal place).