Biologists tagged 180 fish in a lake on January 1. On February​ 1, they returned and collected a random sample of 45 ​fish, 15 of which had been previously tagged. On the basis of this​ experiment, approximately how many fish does the lake​ have?

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To estimate the total number of fish in the lake, we can use a mathematical technique called extrapolation.

Let's start with the proportion of tagged fish in the sample:

Proportion of tagged fish in the sample = Number of tagged fish in the sample / Total number of fish in the sample

We know that 15 out of the 45 fish in the sample were tagged. So, the proportion of tagged fish in the sample is 15/45.

Now, we can use this proportion to estimate the total number of fish in the lake. We assume that the proportion of tagged fish in the sample is representative of the proportion of tagged fish in the entire lake.

If we set up a proportion using this assumption:

Proportion of tagged fish in the sample = Proportion of tagged fish in the entire lake

15/45 = Number of tagged fish in the entire lake / Total number of fish in the entire lake

Since we want to find the total number of fish in the entire lake, we can rearrange the equation:

Total number of fish in the entire lake = (Number of tagged fish in the entire lake / Proportion of tagged fish in the entire lake) × Proportion of tagged fish in the sample

Given that we initially tagged 180 fish in the lake on January 1, we can substitute the values into the equation:

Total number of fish in the entire lake = (180 / Proportion of tagged fish in the entire lake) × (15/45)

Now, to find the approximate number of fish in the lake, we need to know the proportion of tagged fish in the entire lake. However, this information is not provided in the question. Without this value, we cannot calculate the exact number of fish in the lake.

If you have the proportion of tagged fish in the entire lake, you can substitute it into the equation to estimate the total number of fish in the lake.