Question

a bacteria culture initially contains 2500 bacteria and doubles every half hour.
find the size of the bacterial population after 80 minutes.
find the size of the bacterial population after 10 hours.

Explanation:

Step1: Determine the general formula for population growth

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, \( P_0 = 2500 \) and \( T=30 \) minutes.

Step2: Calculate the population after 80 minutes

Substitute \( t = 80 \) into the formula:
\( P(80)=2500\cdot2^{\frac{80}{30}}=2500\cdot2^{\frac{8}{3}} \)
\( 2^{\frac{8}{3}}=(2^{\frac{1}{3}})^8\approx1.26\times1.26\times1.26\times1.26\times1.26\times1.26\times1.26\times1.26\approx10.08 \)
\( P(80)=2500\times10.08 = 25200 \)

Step3: Calculate the population after 10 hours (600 minutes)

Substitute \( t = 600 \) into the formula:
\( P(600)=2500\cdot2^{\frac{600}{30}}=2500\cdot2^{20} \)
\( 2^{20}=1048576 \)
\( P(600)=2500\times1048576=2621440000 \)

Answer:

After 80 minutes: \( 25200 \)
After 10 hours: \( 2621440000 \)