Question

a culture of bacteria has an initial population of 770 bacteria and doubles every 6 hours. using the formula $p_t = p_0 cdot 2^{\frac{t}{d}}$, 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, what is the population of bacteria in the culture after 11 hours, to the nearest whole number?

Explanation:

Step1: Identify given values

We know that \( P_0 = 770 \), \( d = 6 \) hours, and \( t = 11 \) hours. The formula is \( P_t = P_0 \cdot 2^{\frac{t}{d}} \).

Step2: Substitute values into the formula

Substitute \( P_0 = 770 \), \( t = 11 \), and \( d = 6 \) into the formula: \( P_{11} = 770 \cdot 2^{\frac{11}{6}} \).

Step3: Calculate the exponent

First, calculate \( \frac{11}{6} \approx 1.8333 \).

Step4: Calculate the power of 2

Then, calculate \( 2^{1.8333} \approx 2^{1 + \frac{5}{6}} = 2^1 \cdot 2^{\frac{5}{6}} \approx 2 \cdot \sqrt[6]{32} \approx 2 \cdot 2.07 \approx 4.14 \) (alternatively, use a calculator to find \( 2^{11/6} \approx 3.527 \) (more accurately, \( 2^{11/6}=2^{1 + 5/6}=2\times2^{5/6}\approx2\times1.763\approx3.526 \))).

Step5: Multiply by initial population

Now, multiply by 770: \( 770 \times 3.526 \approx 770\times3.526 = 770\times(3 + 0.526)=2310 + 405.02 = 2715.02 \).

Answer:

The population of bacteria after 11 hours is approximately \(\boxed{2715}\) (rounded to the nearest whole number).