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, and [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 :

We are given the formula for exponential growth:

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

where:
- [tex]$P_0$[/tex] is the initial population,
- [tex]$t$[/tex] is the time in hours,
- [tex]$d$[/tex] is the doubling time,
- [tex]$P_t$[/tex] is the population after [tex]$t$[/tex] hours.

For this problem:
- The initial population is [tex]$P_0 = 610$[/tex] bacteria.
- The doubling time is [tex]$d = 6$[/tex] hours.
- The elapsed time is [tex]$t = 13$[/tex] hours.

Step 1: Calculate the exponent

We first compute the exponent [tex]$\frac{t}{d}$[/tex]:

[tex]$$
\frac{t}{d} = \frac{13}{6} \approx 2.16667.
$$[/tex]

Step 2: Substitute into the formula

Substitute the given values into the exponential growth formula:

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

Step 3: Evaluate the expression

Evaluating [tex]$2^{\frac{13}{6}}$[/tex] gives a value, and multiplying that result by 610 yields approximately:

[tex]$$
P_t \approx 2738.80739787487.
$$[/tex]

Step 4: Round to the nearest whole number

Rounding the result to the nearest whole number, we have:

[tex]$$
P_t \approx 2739.
$$[/tex]

Final Answer:

After 13 hours, the population of bacteria is approximately [tex]$2739$[/tex].

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