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Answer :
We start with the formula for exponential growth:
[tex]$$
P(t) = P_0 \cdot 2^{\frac{t}{d}},
$$[/tex]
where:
- [tex]$P_0 = 43000$[/tex] is the initial population,
- [tex]$t = 13$[/tex] hours is the time elapsed,
- [tex]$d = 5$[/tex] hours is the doubling time.
Step 1: Compute the exponent
The exponent is given by [tex]$\frac{t}{d}$[/tex]:
[tex]$$
\frac{t}{d} = \frac{13}{5} = 2.6.
$$[/tex]
Step 2: Substitute into the formula
Substitute the values into the growth formula:
[tex]$$
P(13) = 43000 \cdot 2^{2.6}.
$$[/tex]
Step 3: Evaluate the expression
Evaluating [tex]$2^{2.6}$[/tex] gives a certain value which, when multiplied by [tex]$43000$[/tex], results in an approximate population of [tex]$260703.2494397885$[/tex] bacteria.
Step 4: Round to the nearest whole number
Rounding the computed population to the nearest whole number, we obtain:
[tex]$$
P(13) \approx 260703.
$$[/tex]
Thus, the population of the bacteria culture after 13 hours is approximately [tex]$260703$[/tex] bacteria.
[tex]$$
P(t) = P_0 \cdot 2^{\frac{t}{d}},
$$[/tex]
where:
- [tex]$P_0 = 43000$[/tex] is the initial population,
- [tex]$t = 13$[/tex] hours is the time elapsed,
- [tex]$d = 5$[/tex] hours is the doubling time.
Step 1: Compute the exponent
The exponent is given by [tex]$\frac{t}{d}$[/tex]:
[tex]$$
\frac{t}{d} = \frac{13}{5} = 2.6.
$$[/tex]
Step 2: Substitute into the formula
Substitute the values into the growth formula:
[tex]$$
P(13) = 43000 \cdot 2^{2.6}.
$$[/tex]
Step 3: Evaluate the expression
Evaluating [tex]$2^{2.6}$[/tex] gives a certain value which, when multiplied by [tex]$43000$[/tex], results in an approximate population of [tex]$260703.2494397885$[/tex] bacteria.
Step 4: Round to the nearest whole number
Rounding the computed population to the nearest whole number, we obtain:
[tex]$$
P(13) \approx 260703.
$$[/tex]
Thus, the population of the bacteria culture after 13 hours is approximately [tex]$260703$[/tex] bacteria.
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