In a lab experiment, $6300$ bacteria are placed in a petri dish. The conditions are such that the number of bacteria is able to double every $8$ hours. How many bacteria would there be after $19$ hours, to the nearest whole number? $\newline$ Answer:

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Q. In a lab experiment,

6300

bacteria are placed in a petri dish. The conditions are such that the number of bacteria is able to double every

8

hours. How many bacteria would there be after

19

hours, to the nearest whole number?

\newline

Answer:

Calculate Doubling Periods: Determine the number of times the bacteria population will double in $19$ hours. $\newline$ Since the bacteria double every $8$ hours, we divide the total time by the doubling period. $\newline$ $19$ hours $\div$ $8$ hours per doubling period = $2.375$ doubling periods.
Calculate Bacteria After Doubling Periods: Since bacteria can only double a whole number of times, we need to consider only the whole number part of the doubling periods for the calculation. $\newline$ The number of complete doubling periods in $19$ hours is $2$ (since we cannot have a fraction of a doubling period).
Calculate Remaining Time: Calculate the number of bacteria after the complete doubling periods. $\newline$ The population doubles $2$ times, so we multiply the initial number of bacteria by $2$ raised to the power of the number of doublings. $\newline$ $6300$ bacteria $\times (2^2) = 6300 \times 4 = 25200$ bacteria after $16$ hours ( $2$ complete doubling periods).
Calculate Growth During Remaining Time: Determine the remaining time after the last complete doubling period. $19$ hours - $16$ hours ( $2$ complete doubling periods of $8$ hours each) = $3$ hours remaining.
Apply Growth for Partial Period: Calculate the growth of bacteria during the remaining $3$ hours. $\newline$ Since the bacteria double every $8$ hours, we can find the fraction of doubling that occurs in $3$ hours by dividing $3$ by $8$ . $\newline$ $3$ hours $\div$ $8$ hours per doubling period = $0.375$ of a doubling period.
Round Bacteria Count: Apply the growth for the partial doubling period to the bacteria count after $16$ hours. $\newline$ We need to multiply the number of bacteria after $16$ hours by $2$ raised to the power of $0.375$ to find the number of bacteria after the remaining $3$ hours. $\newline$ $25200$ bacteria $\times (2^{0.375}) \approx 25200 \times 1.333 \approx 33600$ bacteria after $19$ hours.
Round Bacteria Count: Apply the growth for the partial doubling period to the bacteria count after $16$ hours. $\newline$ We need to multiply the number of bacteria after $16$ hours by $2$ raised to the power of $0.375$ to find the number of bacteria after the remaining $3$ hours. $\newline$ $25200$ bacteria $\times (2^{0.375}) \approx 25200 \times 1.333 \approx 33600$ bacteria after $19$ hours.Round the result to the nearest whole number, as bacteria count cannot be a fraction. $\newline$ The number of bacteria after $19$ hours, rounded to the nearest whole number, is approximately $33600$ .