In a lab experiment, $30$ bacteria are placed in a petri dish. The conditions are such that the number of bacteria is able to double every $7$ hours. How many bacteria would there be after $16$ hours, to the nearest whole number? $\newline$ Answer:

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Q. In a lab experiment,

30

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

7

hours. How many bacteria would there be after

16

hours, to the nearest whole number?

\newline

Answer:

Calculate Doublings: Determine the number of times the bacteria will double in $16$ hours. $\newline$ Since the bacteria double every $7$ hours, we divide the total time by the doubling period. $\newline$ $16$ hours $\div$ $7$ hours per doubling period $= 2.2857$ doublings $\newline$ However, since bacteria can only double a whole number of times, we need to consider only complete doublings within the $16$ -hour period. $\newline$ $2$ complete doublings will occur in $14$ hours ( $2 \times 7$ hours).
Calculate Bacteria After Doublings: Calculate the number of bacteria after the complete doublings. $\newline$ The initial number of bacteria is $30$ . After each doubling, the number of bacteria will multiply by $2$ . $\newline$ After $2$ doublings: $30 \times 2 \times 2 = 30 \times 4 = 120$ bacteria.
Remaining Time: Determine the remaining time after the last complete doubling. $16$ hours - $14$ hours = $2$ hours remaining.
Calculate Growth in $2$ Hours: Calculate the growth of bacteria during the remaining $2$ hours. $\newline$ Since the bacteria double every $7$ hours, we need to find out how much they grow in $2$ hours. $\newline$ Growth in $2$ hours = $2$ hours / $7$ hours per doubling period. $\newline$ This is a fraction of a doubling period, so we need to raise $2$ (the doubling factor) to the power of this fraction. $\newline$ Growth factor = $2^{(2/7)}$ .
Calculate Bacteria After $16$ Hours: Calculate the number of bacteria after $16$ hours. $\newline$ We multiply the number of bacteria after the complete doublings by the growth factor. $\newline$ Number of bacteria after $16$ hours $= 120 \times 2^{\frac{2}{7}}$ . $\newline$ Using a calculator, $2^{\frac{2}{7}} \approx 1.24573$ . $\newline$ Number of bacteria after $16$ hours $\approx 120 \times 1.24573 \approx 149.4876$ .
Round to Nearest Whole Number: Round the number of bacteria to the nearest whole number. $\newline$ Since we cannot have a fraction of a bacterium, we round to the nearest whole number. $\newline$ Approximately $149.4876$ rounds to $149$ bacteria.