The bacteria in a laboratory culture increased from an initi

The bacteria in a laboratory culture increased from an initial population of 500 to 1,500 in 3 hours.How long will it take the population to reach 30,000? (Give your answer correct to the nearest whole number.)

Solution

Let the bacterial growth model be A = A₀e^kt , where A₀ is the initial number, A is the number after time t and k is the constant of growth. Here, A₀ = 500 and A = 1500 after 3 hours. Then 1500 = 500e^3k or, e^3k = 1500/500 = 3. On taking natural log of both the sides, we get 3k ln e = ln 3 or, 3k = 1.098612289 so that k = 1.098612289/3 = 0.366204096. Thus A = 500e^.366204096t.

Let the no. of bacteria reach 30000 after t hours. Then 30000 = 500e^.366204096t or, e^0.366204096t   = 30000/500 = 60.Now, on taking natural log of both the sides, we get 0. 366204096t = ln 60 = 4.094344562. Therefore, t = 4.094344562/0. 366204096 = 11.18 hours, say 11 hours ( on rounding off to the nearest whole number.