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$E. coli$ is among the fastest-growing bacteria, with a generation (or doubling) time of $20$ minutes under optimal conditions. After $60$ minutes, the number of bacteria in a culture of $E. coli$ was $400$ . Approximately how many bacteria were in the culture after $30$ minutes?

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Exponential growth and decay: word problems

Full solution

Q.

E. coli

is among the fastest-growing bacteria, with a generation (or doubling) time of

20

minutes under optimal conditions. After

60

minutes, the number of bacteria in a culture of

E. coli

was

400

. Approximately how many bacteria were in the culture after

30

minutes?

Calculate Doublings: We know that extit{E. coli} doubles every $20$ minutes. After $60$ minutes, the number of bacteria is $400$ . Let's find out how many times the bacteria have doubled in $60$ minutes. $\newline$ $60$ minutes $/$ $20$ minutes per doubling $=$ $3$ doublings.
Find Bacteria After $40$ Minutes: Since the bacteria double every $20$ minutes, we can work backwards to find the number of bacteria after $30$ minutes. After $60$ minutes, there are $400$ bacteria, which means after $40$ minutes (one doubling period earlier), there would be half of that number. $\newline$ $400$ bacteria $/$ $2$ = $200$ bacteria after $40$ minutes.
Find Bacteria After $30$ Minutes: Now, we need to go back one more doubling period to find the number of bacteria after $30$ minutes. Since $40$ minutes is one doubling period after $30$ minutes, we halve the number of bacteria again. $\newline$ $200$ bacteria / $2$ = $100$ bacteria after $30$ minutes.

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