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There is a population of $30$ bacteria in a colony. If the number of bacteria doubles every $500$ hours, what will the population be $1,000$ hours from now?

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Exponential growth and decay: word problems

Full solution

Q. There is a population of

30

bacteria in a colony. If the number of bacteria doubles every

500

hours, what will the population be

1,000

hours from now?

Identify Initial Population: Determine the initial population and the doubling time. Initial population $a$ = $30$ bacteria, Doubling time $T$ = $500$ hours.
Calculate Doubling Periods: Calculate the number of doubling periods in $1,000$ hours. $\newline$ Total time $(t) = 1,000$ hours. $\newline$ Number of doubling periods $(x) = \frac{t}{T} = \frac{1,000}{500} = 2$ .
Apply Exponential Growth Formula: Apply the formula for exponential growth: $P(x) = a(b)^x$ . $\newline$ Here, $b = 2$ (since the population doubles), and $x = 2$ (from previous step). $\newline$ $P(x) = 30(2)^2 = 30 \times 4 = 120$ .

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