High School

A culture of bacteria has an initial population of 610 bacteria and doubles every 6 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, we can use 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 in hours.

In this scenario:
- The initial population, [tex]\( P_0 \)[/tex], is 610 bacteria.
- The time, [tex]\( t \)[/tex], we are checking for is 13 hours.
- The doubling time, [tex]\( d \)[/tex], is 6 hours.

Now, plug these values into the formula:

[tex]\[ P_t = 610 \cdot 2^{\frac{13}{6}} \][/tex]

First, calculate the exponent:

[tex]\[ \frac{13}{6} = 2.1667 \][/tex]

Now, calculate [tex]\( 2^{2.1667} \)[/tex]:

[tex]\[ 2^{2.1667} \approx 3.922 \][/tex]

Finally, multiply this result by the initial population:

[tex]\[ P_t = 610 \cdot 3.922 \approx 2390.807 \][/tex]

Round this result to the nearest whole number to get the population after 13 hours:

The estimated population of bacteria after 13 hours is approximately 2739 bacteria.

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