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Answer :
After 13 hours, the population of bacteria in the culture is approximately 19,372, rounded to the nearest whole number.
To find the population of bacteria after 13 hours using the given formula [tex]\(P_t = P_0 \cdot 2^{\frac{t}{d}}\),[/tex] where [tex]\(P_t\)[/tex] is the population after [tex]\(t\) hours, \(P_0\)[/tex] is the initial population, and \(d\) is the doubling time, we first need to determine the doubling time.
The population doubles every 5 hours, so [tex]\(d = 5\)[/tex] hours. The initial population (\(P_0\)) is 2900 bacteria. Substituting these values into the formula, we get:
[tex]\[P_{13} = 2900 \times 2^{\frac{13}{5}}\][/tex]
Simplifying the exponent:
[tex]\[P_{13} = 2900 \times 2^{2.6}\][/tex]
Calculating [tex]\(2^{2.6}\)[/tex] gives approximately 6.68. Multiplying this by 2900:
[tex]\[P_{13} \approx 2900 \times 6.68 \approx 19372\][/tex]
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