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Answer :
We start with the formula that models exponential growth:
[tex]$$
P_t = P_0 \cdot 2^{\frac{t}{d}},
$$[/tex]
where
- [tex]$P_0 = 790$[/tex] is the initial population,
- [tex]$t = 14$[/tex] hours is the elapsed time,
- [tex]$d = 5$[/tex] hours is the doubling time.
Step 1: Calculate the exponent
The exponent in the formula is given by [tex]$\frac{t}{d}$[/tex]:
[tex]$$
\frac{t}{d} = \frac{14}{5} = 2.8.
$$[/tex]
Step 2: Evaluate the growth factor
Next, we evaluate the power of [tex]$2$[/tex]:
[tex]$$
2^{\frac{14}{5}} = 2^{2.8}.
$$[/tex]
Though the exact calculation of [tex]$2^{2.8}$[/tex] is complex without a calculator, it is approximately [tex]$6.9654$[/tex]. (This value is the result of [tex]$2^{2.8}$[/tex] computed accurately.)
Step 3: Compute the population
Now, substitute the values into the formula:
[tex]$$
P_{14} = 790 \cdot 2^{2.8}.
$$[/tex]
Multiplying these together:
[tex]$$
P_{14} \approx 790 \cdot 6.9654 \approx 5501.88.
$$[/tex]
Step 4: Round to the nearest whole number
Since the population must be a whole number, we round [tex]$5501.88$[/tex] to get:
[tex]$$
P_{14} \approx 5502.
$$[/tex]
Thus, after 14 hours, the population of bacteria is approximately [tex]$\boxed{5502}$[/tex] bacteria.
[tex]$$
P_t = P_0 \cdot 2^{\frac{t}{d}},
$$[/tex]
where
- [tex]$P_0 = 790$[/tex] is the initial population,
- [tex]$t = 14$[/tex] hours is the elapsed time,
- [tex]$d = 5$[/tex] hours is the doubling time.
Step 1: Calculate the exponent
The exponent in the formula is given by [tex]$\frac{t}{d}$[/tex]:
[tex]$$
\frac{t}{d} = \frac{14}{5} = 2.8.
$$[/tex]
Step 2: Evaluate the growth factor
Next, we evaluate the power of [tex]$2$[/tex]:
[tex]$$
2^{\frac{14}{5}} = 2^{2.8}.
$$[/tex]
Though the exact calculation of [tex]$2^{2.8}$[/tex] is complex without a calculator, it is approximately [tex]$6.9654$[/tex]. (This value is the result of [tex]$2^{2.8}$[/tex] computed accurately.)
Step 3: Compute the population
Now, substitute the values into the formula:
[tex]$$
P_{14} = 790 \cdot 2^{2.8}.
$$[/tex]
Multiplying these together:
[tex]$$
P_{14} \approx 790 \cdot 6.9654 \approx 5501.88.
$$[/tex]
Step 4: Round to the nearest whole number
Since the population must be a whole number, we round [tex]$5501.88$[/tex] to get:
[tex]$$
P_{14} \approx 5502.
$$[/tex]
Thus, after 14 hours, the population of bacteria is approximately [tex]$\boxed{5502}$[/tex] bacteria.
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