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A culture of bacteria has an initial population of 65,000 bacteria and doubles every 2 hours.

Using the formula [tex]$P_t = P_0 \cdot 2^{\frac{t}{d}}$[/tex], where:
- [tex]$P_t$[/tex] is the population after [tex]$t$[/tex] hours,
- [tex]$P_0$[/tex] is the initial population,
- [tex]$t$[/tex] is the time in hours,
- [tex]$d$[/tex] is the doubling time,

What is the population of bacteria in the culture after 13 hours, to the nearest whole number?

Answer :

To find the population of bacteria after 13 hours when the initial population is 65,000 and it doubles every 2 hours, we can use the exponential growth formula:

[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]

Here’s a step-by-step breakdown:

1. Identify the Variables:
- [tex]\( P_0 \)[/tex]: Initial population, which is 65,000.
- [tex]\( t \)[/tex]: Time after which we want to find the population, which is 13 hours.
- [tex]\( d \)[/tex]: Doubling time, which is 2 hours.

2. Plug in the Values:
[tex]\[ P_t = 65000 \cdot 2^{\frac{13}{2}} \][/tex]

3. Calculate the Exponent:
[tex]\[ \frac{13}{2} = 6.5 \][/tex]

4. Calculate [tex]\( 2^{6.5} \)[/tex]:
To find [tex]\( 2^{6.5} \)[/tex], calculate the approximate value which is roughly 90.5097 (this involves using a calculator for precise values of powers of 2).

5. Compute [tex]\( P_t \)[/tex]:
[tex]\[ P_t = 65000 \cdot 90.5097 \][/tex]

6. Multiply to Find the Result:
[tex]\[ P_t \approx 5883128.419472076 \][/tex]

7. Round to the Nearest Whole Number:
The population of bacteria after 13 hours, rounded to the nearest whole number, is [tex]\( 5,883,128 \)[/tex].

So, the population of bacteria after 13 hours is approximately 5,883,128 bacteria.

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