In a lab experiment, $480$ bacteria are placed in a petri dish. The conditions are such that the number of bacteria is able to double every $22$ hours. How many bacteria would there be after $25$ hours, to the nearest whole number? $\newline$ Answer:

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Q. In a lab experiment,

480

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

22

hours. How many bacteria would there be after

25

hours, to the nearest whole number?

\newline

Answer:

Calculate Doubling Periods: We need to determine how many times the bacteria population will double in $25$ hours, given that it doubles every $22$ hours. To do this, we divide the total time by the doubling period. $\newline$ Calculation: $25$ hours $/$ $22$ hours per doubling $=$ approximately $1.13636$ doublings.
Determine Growth Factor: Since we can't have a fraction of a doubling, we need to consider that the bacteria will have doubled once in the $25$ -hour period, and we will have some growth towards the next doubling. To find the growth factor for the remaining time, we take the fraction of the doubling period that has passed after one full doubling. $\newline$ Calculation: $25$ hours $-$ $22$ hours $=$ $3$ hours. This is the time towards the next doubling. $\newline$ Growth factor for $3$ hours $=$ $3$ hours $/$ $22$ hours $=$ approximately $25$ $2$ of a doubling period.
Calculate Bacteria After One Doubling: To find the number of bacteria after one full doubling and additional growth, we first calculate the number of bacteria after one doubling, and then apply the growth factor for the additional $3$ hours. $\newline$ Calculation: After one doubling, the number of bacteria is $480 \times 2 = 960$ . $\newline$ To find the growth for the additional $3$ hours, we need to raise $2$ to the power of the growth factor. $\newline$ Growth for $3$ hours = $2^{0.13636}$ .
Calculate Growth for Additional Time: We calculate the growth for the additional $3$ hours using the growth factor. $\newline$ Calculation: $2^{0.13636} \approx 1.09417$ (using a calculator).
Calculate Total Bacteria After $25$ Hours: Now we multiply the number of bacteria after one doubling by the growth factor to find the total number of bacteria after $25$ hours. $\newline$ Calculation: $960 \times 1.09417 \approx 1050.4032$ .
Round to Nearest Whole Number: We round the result to the nearest whole number, as the question asks for the number of bacteria to the nearest whole number. $\newline$ Calculation: $1050.4032$ rounded to the nearest whole number is $1050$ .