High School

A culture of bacteria has an initial population of 460 bacteria and doubles every 4 hours. Using the formula [tex]P_t = P_0 \cdot 2^{t/d}[/tex], where [tex]P_t[/tex] is the population after [tex]t[/tex] hours, [tex]P_0[/tex] is the initial population, [tex]t[/tex] is the time in hours, and [tex]d[/tex] is the doubling time, what is the population of bacteria in the culture after 11 hours, to the nearest whole number?

Answer :

After 11 hours, the population of bacteria in the culture is approximately 2603, rounded to the nearest whole number, using the given growth formula.

To find the population of bacteria in the culture after 11 hours, we can use the given formula:

[tex]\[ P_t = P_0 \times 2^{t/d} \][/tex]

Given:

- Initial population [tex](\( P_0 \))[/tex] = 460 bacteria

- Doubling time [tex](\( d \))[/tex] = 4 hours

- Time [tex](\( t \))[/tex]= 11 hours

Substituting these values into the formula:

[tex]\[ P_{11} = 460 \times 2^{11/4} \][/tex]

Now, let's calculate:

[tex]\[ P_{11} = 460 \times 2^{2.75} \][/tex]

Using a calculator:

[tex]\[ P_{11} \approx 460 \times 5.6568 \][/tex]

[tex]\[ P_{11} \approx 2602.768 \][/tex]

Rounding to the nearest whole number:

[tex]\[ P_{11} \approx 2603 \][/tex]

So, the population of bacteria in the culture after 11 hours is approximately 2603 bacteria.

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