A culture of bacteria has an initial population of 83000 bacteria and doubles every 10 hours. Using the formula P, start subscript, t, end subscript, equals, P, start subscript, 0, end subscript, dot, 2, start superscript, start fraction, t, divided by, d, end fraction, end superscriptP t =P 0 ⋅2 dt , where P, start subscript, t, end subscriptP t is the population after t hours, P, start subscript, 0, end subscriptP 0 is the initial population, t is the time in hours and d is the doubling time, what is the population of bacteria in the culture after 3 hours, to the nearest whole number?
Question
A culture of bacteria has an initial population of 83000 bacteria and doubles every 10 hours. Using the formula P, start subscript, t, end subscript, equals, P, start subscript, 0, end subscript, dot, 2, start superscript, start fraction, t, divided by, d, end fraction, end superscriptP t =P 0 ⋅2 dt , where P, start subscript, t, end subscriptP t is the population after t hours, P, start subscript, 0, end subscriptP 0 is the initial population, t is the time in hours and d is the doubling time, what is the population of bacteria in the culture after 3 hours, to the nearest whole number?
Solution
To solve this problem, we will use the formula given:
P_t = P_0 * 2^(t/d)
where:
- P_t is the population after t hours,
- P_0 is the initial population,
- t is the time in hours,
- d is the doubling time.
Given in the problem:
- P_0 = 83000 (initial population),
- t = 3 hours (time after which we want to find the population),
- d = 10 hours (doubling time).
Substituting these values into the formula, we get:
P_t = 83000 * 2^(3/10)
To simplify the exponent, divide 3 by 10 to get 0.3:
P_t = 83000 * 2^0.3
Now, calculate 2^0.3. You can use a calculator for this, and you should get approximately 1.2311444.
P_t = 83000 * 1.2311444
Finally, multiply 83000 by 1.2311444 to get the population after 3 hours. This should give you approximately 102185.0052.
Since we can't have a fraction of a bacterium, round this to the nearest whole number. The population of bacteria in the culture after 3 hours is therefore approximately 102185.
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