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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, and
- [tex]$d$[/tex] is the doubling time.
For this problem:
- [tex]$P_0 = 790$[/tex],
- [tex]$t = 14$[/tex] hours, and
- [tex]$d = 5$[/tex] hours.
Step 1: Compute the exponent.
The exponent is calculated by
[tex]$$
\frac{t}{d} = \frac{14}{5} = 2.8.
$$[/tex]
Step 2: Substitute into the growth formula.
We substitute the values into the formula:
[tex]$$
P_{14} = 790 \cdot 2^{2.8}.
$$[/tex]
Step 3: Calculate the population and round to the nearest whole number.
Evaluating the expression, we find
[tex]$$
P_{14} \approx 790 \cdot 2^{2.8} \approx 5501.88.
$$[/tex]
Rounding [tex]$5501.88$[/tex] to the nearest whole number gives
[tex]$$
P_{14} \approx 5502.
$$[/tex]
Thus, the population of bacteria after 14 hours is approximately [tex]$\boxed{5502}$[/tex] bacteria.
[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, and
- [tex]$d$[/tex] is the doubling time.
For this problem:
- [tex]$P_0 = 790$[/tex],
- [tex]$t = 14$[/tex] hours, and
- [tex]$d = 5$[/tex] hours.
Step 1: Compute the exponent.
The exponent is calculated by
[tex]$$
\frac{t}{d} = \frac{14}{5} = 2.8.
$$[/tex]
Step 2: Substitute into the growth formula.
We substitute the values into the formula:
[tex]$$
P_{14} = 790 \cdot 2^{2.8}.
$$[/tex]
Step 3: Calculate the population and round to the nearest whole number.
Evaluating the expression, we find
[tex]$$
P_{14} \approx 790 \cdot 2^{2.8} \approx 5501.88.
$$[/tex]
Rounding [tex]$5501.88$[/tex] to the nearest whole number gives
[tex]$$
P_{14} \approx 5502.
$$[/tex]
Thus, the population of bacteria after 14 hours is approximately [tex]$\boxed{5502}$[/tex] bacteria.
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