To determine the population of the bacteria after 516 minutes, we first need to calculate how many doubling periods fit into that time frame.
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Find the number of doubling periods: \[ \text{Number of doublings} = \frac{\text{Total time}}{\text{Doubling time}} = \frac{516 \text{ minutes}}{129 \text{ minutes}} = 4 \]
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Since the population doubles every doubling period, we can calculate the population after 4 doublings using the formula: \[ \text{New population} = \text{Initial population} \times 2^{\text{Number of doublings}} \] Plugging in the values: \[ \text{New population} = 10,710 \times 2^4 \]
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Calculate \(2^4\): \[ 2^4 = 16 \]
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Now multiply the initial population by the growth factor: \[ \text{New population} = 10,710 \times 16 = 171,360 \]
Therefore, the population of the bacteria after 516 minutes will be 171,360.