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A tilapia fish hatchery selectively releases fish when the populations have increased beyond a certain target level. In order to estimate the current fish population, workers at the hatchery catch $780$ fish and mark them with special paint. Then a little while later, they catch $1,200$ fish, among which $72$ are marked. To the nearest whole number, what is the best estimate for the fish population?

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Estimate population size using proportions

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Q. A tilapia fish hatchery selectively releases fish when the populations have increased beyond a certain target level. In order to estimate the current fish population, workers at the hatchery catch

780

fish and mark them with special paint. Then a little while later, they catch

1,200

fish, among which

72

are marked. To the nearest whole number, what is the best estimate for the fish population?

Set up proportion: Step $1$ : Set up the proportion based on the capture-recapture method. $\newline$ We know: $\newline$ - Marked fish found: $72$ $\newline$ - Total fish caught in second sample: $1200$ $\newline$ - Total fish marked initially: $780$ $\newline$ Let $p$ be the estimated fish population. $\newline$ Set up the proportion: $\newline$ $\frac{72}{1200} = \frac{780}{p}$
Solve proportion: Step $2$ : Solve the proportion by cross-multiplying to find $p$ . $\newline$ Cross multiply: $\newline$ $72 \times p = 780 \times 1200$ $\newline$ $72 \times p = 936000$
Isolate p: Step $3$ : Divide both sides by $72$ to isolate $p$ . $\newline$ $p = \frac{936000}{72}$ $\newline$ $p = 13000$

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