A culture of bacteria has an initial population of 7800 bacteria and doubles every 7 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 20 hours, to the nearest whole number?
Question
A culture of bacteria has an initial population of 7800 bacteria and doubles every 7 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 20 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 = 7800 (the initial population),
- t = 20 hours, and
- d = 7 hours (the doubling time).
Substituting these values into the formula, we get:
P_t = 7800 * 2^(20/7)
To simplify the exponent, divide 20 by 7:
20 ÷ 7 ≈ 2.857
So the formula becomes:
P_t = 7800 * 2^2.857
Now, calculate 2^2.857. This is approximately 7.389.
So the formula becomes:
P_t = 7800 * 7.389
Finally, multiply 7800 by 7.389 to find the population after 20 hours:
P_t ≈ 57632.2
Rounding to the nearest whole number, the population of bacteria in the culture after 20 hours is approximately 57632.
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