Question

the count in a bacteria culture was initially 700, and after 40 minutes the population had increased to 1300.

find the doubling period.

find the population after 80 minutes.

when will the population reach 14000?

Explanation:

Step1: Write the growth model

$$P(t) = P_0 \cdot e^{kt}$$

Step2: Substitute known values

$$1300 = 700 \cdot e^{k \cdot 40}$$

Step3: Solve for growth rate

$$k = \frac{\ln(13/7)}{40} \approx 0.01548$$

Step4: Calculate doubling period

$$d = \frac{\ln(2)}{k} \approx 44.76$$

Step5: Calculate population at 80 minutes

$$P(80) = 700 \cdot e^{0.01548 \cdot 80} \approx 2414$$

Step6: Set up target population equation

$$14000 = 700 \cdot e^{0.01548 \cdot t}$$

Step7: Solve for target time

$$t = \frac{\ln(20)}{0.01548} \approx 193.33$$

Answer:

The doubling period is approximately \(44.76\) minutes.
The population after 80 minutes is approximately \(2414\).
The population will reach 14000 after approximately \(193.33\) minutes.