A culture of bacteria has an initial population of 9100 bacteria and doubles every 4 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 5 hours, to the nearest whole number?
Question
A culture of bacteria has an initial population of 9100 bacteria and doubles every 4 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 5 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, and
- d is the doubling time.
We know that:
- P_0 = 9100 (the initial population),
- t = 5 hours (the time we want to find the population for), and
- d = 4 hours (the doubling time).
Substituting these values into the formula, we get:
P_t = 9100 * 2^(5/4)
To simplify the exponent, 5 divided by 4 equals 1.25. So the formula becomes:
P_t = 9100 * 2^1.25
Now, we just need to calculate the value of 2^1.25 and multiply it by 9100 to get the population after 5 hours.
2^1.25 is approximately 2.378414230005442, and multiplying this by 9100 gives us:
P_t ≈ 9100 * 2.378414230005442 ≈ 21633.57
Rounding to the nearest whole number, the population of bacteria in the culture after 5 hours is approximately 21634.
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