Question

a bacteria culture initially contains 3000 bacteria and doubles every half hour.
find the size of the baterial population after 80 minutes.

find the size of the baterial population after 4 hours.

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Explanation:

Step1: Determine the general formula

The formula for exponential growth is \( P(t)=P_0\cdot2^{\frac{t}{T}} \), where \( P_0 \) is the initial population, \( t \) is the time elapsed, and \( T \) is the time it takes for the population to double. Here, \( P_0 = 3000\) and \( T=30\) minutes.

Step2: Calculate for \( t = 80 \) minutes

Substitute \( P_0 = 3000\), \( t = 80\), and \( T = 30\) into the formula:
\( P(80)=3000\cdot2^{\frac{80}{30}}=3000\cdot2^{\frac{8}{3}}\)
\( 2^{\frac{8}{3}}=\sqrt[3]{2^8}=\sqrt[3]{256}\approx6.3496\)
\( P(80)\approx3000\times6.3496 = 19048.8\approx19049\)

Step3: Calculate for \( t = 4\) hours (since \( 4\) hours \(= 240\) minutes)

Substitute \( P_0 = 3000\), \( t = 240\), and \( T = 30\) into the formula:
\( P(240)=3000\cdot2^{\frac{240}{30}}=3000\cdot2^8\)
Since \( 2^8=256\)
\( P(240)=3000\times256 = 768000\)

Answer:

• After 80 minutes: \( 19049\)
• After 4 hours: \( 768000\)