The number of bacteria in a certain population increases according to a continuous exponential growth model, with a growth rate parameter of per hour.
How many hours does it take for the size of the sample to double?
Note: This is a continuous exponential growth model.
Do not round any intermediate computations, and round your answer to the nearest hundredth.

Question

The number of bacteria in a certain population increases according to a continuous exponential growth model, with a growth rate parameter of

per hour.
How many hours does it take for the size of the sample to double?
Note: This is a continuous exponential growth model.
Do not round any intermediate computations, and round your answer to the nearest hundredth.

UpStudy AI · Accepted Answer

1. The continuous exponential growth model is given by  
   $$
   N(t) = N_0 e^{rt},
   $$  
   where $$ N_0 $$ is the initial number of bacteria, $$ r $$ is the growth rate per hour, and $$ t $$ is time in hours.

2. We are given that $$ r = 1.3\% = 0.013 $$. The population doubles when $$ N(t) = 2N_0 $$. Thus,  
   $$
   2N_0 = N_0 e^{0.013t}.
   $$

3. Dividing both sides by $$ N_0 $$ (assuming $$ N_0 > 0 $$) gives  
   $$
   2 = e^{0.013t}.
   $$

4. Taking the natural logarithm of both sides, we have  
   $$
   \ln(2) = 0.013t.
   $$

5. Solving for $$ t $$, we obtain  
   $$
   t = \frac{\ln(2)}{0.013}.
   $$

6. Calculating with no intermediate rounding:  
   $$
   \ln(2) \approx 0.69314718056,
   $$  
   so  
   $$
   t \approx \frac{0.69314718056}{0.013} \approx 53.31824773.
   $$

7. Rounding to the nearest hundredth, the time required is  
   $$
   t \approx 53.32 \text{ hours}.
   $$
It takes approximately 53.32 hours for the bacteria population to double.

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