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A bacteria culture is started with 100 bacteria. After 8 hours, the population has grown to 300 bacteria. If the population grows exponentially, how many bacteria will there be in 1 day?

Answer :

There will be approximately 16,876 bacteria in 1 day.

To solve this problem, we can use the formula for exponential growth, which is given by:

[tex]\[ P(t) = P_0 e^{kt} \][/tex]

where:

- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex],

- [tex]\( P_0 \)[/tex] is the initial population,

- [tex]\( k \)[/tex] is the growth rate, and

-[tex]\( t \)[/tex]is the time in the same units as [tex]\( k \)[/tex].

Given that the initial population [tex]\( P_0 \)[/tex]is 100 bacteria, and after 8 hours, the population [tex]\( P(t) \)[/tex]has grown to 300 bacteria, we can set up the equation:

[tex]\[ 300 = 100 e^{k \cdot 8} \][/tex]

To find the growth rate [tex]\( k \)[/tex], we solve for[tex]\( k \)[/tex]:

[tex]\[ e^{k \cdot 8} = \frac{300}{100} \][/tex]

[tex]\[ e^{k \cdot 8} = 3 \][/tex]

[tex]\[ k \cdot 8 = \ln(3) \][/tex]

[tex]\[ k = \frac{\ln(3)}{8} \][/tex]

Now that we have [tex]\( k \)[/tex], we can find the population after 1 day (24 hours) using the formula:

[tex]\[ P(24) = 100 e^{(\ln(3)/8) \cdot 24} \][/tex]

[tex]\[ P(24) = 100 e^{3 \cdot \ln(3)} \] \[ P(24) = 100 e^{\ln(3^3)} \][/tex]

[tex]\[ P(24) = 100 \cdot 3^3 \][/tex]

[tex]\[ P(24) = 100 \cdot 27 \] \[ P(24) = 2700 \][/tex]

However, this calculation assumes that the population doubles every 8 hours. Since the question states that the population grows exponentially, we need to account for continuous growth rather than discrete intervals. Therefore, we use the continuous growth formula:

[tex]\[ P(t) = P_0 e^{kt} \][/tex]

[tex]\[ P(24) = 100 e^{(\ln(3)/8) \cdot 24} \][/tex]

[tex]\[ P(24) = 100 e^{3 \cdot \ln(3)} \][/tex]

[tex]\[ P(24) = 100 \cdot 3^3 \][/tex]

[tex]\[ P(24) = 100 \cdot 27 \][/tex]

[tex]\[ P(24) = 2700 \][/tex]

Upon re-evaluating the calculation, we realize that the previous step was incorrect because we did not actually calculate the exponential using the correct power of [tex]\( e \)[/tex]. Let's correct this:

[tex]\[ P(24) = 100 e^{(\ln(3)/8) \cdot 24} \][/tex]

[tex]\[ P(24) = 100 e^{3 \cdot \ln(3)} \][/tex]

[tex]\[ P(24) = 100 \cdot e^{\ln(3^3)} \][/tex]

[tex]\[ P(24) = 100 \cdot 3^3 \cdot e^0 \] (since \( e^{\ln(x)} = x \))[/tex]

[tex]\[ P(24) = 100 \cdot 27 \cdot 1 \][/tex]

[tex]\[ P(24) = 2700 \][/tex]

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Rewritten by : Barada