In a lab experiment, $10$ bacteria are placed in a petri dish. The conditions are such that the number of bacteria is able to double every $13$ 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,

10

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

13

hours. How many bacteria would there be after

19

hours, to the nearest whole number?

\newline

Answer:

Understand Problem: Understand the problem and determine what is given and what needs to be found. $\newline$ We know: $\newline$ Initial number of bacteria: $10$ $\newline$ Doubling time: $13$ hours $\newline$ Total time elapsed: $19$ hours $\newline$ We need to find the number of bacteria after $19$ hours.
Calculate Doubling Periods: Calculate the number of times the bacteria population will double in $19$ hours. $\newline$ To find the number of doubling periods, we divide the total time by the doubling time. $\newline$ Number of doubling periods $= \frac{\text{Total time elapsed}}{\text{Doubling time}}$ $\newline$ Number of doubling periods $= \frac{19 \text{ hours}}{13 \text{ hours}}$
Find Bacteria After $1$ Period: Perform the division to find the number of doubling periods. $\newline$ Number of doubling periods = $19 \div 13 \approx 1.46$ $\newline$ Since the number of bacteria can only double a whole number of times, we consider only the whole number part of the result, which is $1$ .
Determine Final Bacteria Count: Calculate the number of bacteria after the whole number of doubling periods. $\newline$ After one doubling period, the number of bacteria will be: $\newline$ Number of bacteria after $1$ doubling period = Initial number of bacteria $\times 2^{\text{Number of doubling periods}}$ $\newline$ Number of bacteria after $1$ doubling period = $10 \times 2^1$ $\newline$ Number of bacteria after $1$ doubling period = $10 \times 2$ $\newline$ Number of bacteria after $1$ doubling period = $20$
Determine Final Bacteria Count: Calculate the number of bacteria after the whole number of doubling periods. $\newline$ After one doubling period, the number of bacteria will be: $\newline$ Number of bacteria after $1$ doubling period = Initial number of bacteria $\times 2^{\text{Number of doubling periods}}$ $\newline$ Number of bacteria after $1$ doubling period = $10 \times 2^1$ $\newline$ Number of bacteria after $1$ doubling period = $10 \times 2$ $\newline$ Number of bacteria after $1$ doubling period = $20$ Since the bacteria have not yet reached the second doubling period, we do not double the number again. The number of bacteria after $19$ hours will be approximately the same as after the first doubling period. $\newline$ Final number of bacteria $\approx$ Number of bacteria after $1$ doubling period $\newline$ Final number of bacteria $\approx 20$