As part of a conservation effort for a lake, 70 fish were caught, tagged, and then released. Later, 80 fish were caught from the lake. Four of these 80 fish were found to have tags. Estimate the number of fish in the lake.
4/80 = 5% of the caught fish were tagged.
So, the 70 tagged fish must be about 5% of the total population.
1400 Fish
To estimate the number of fish in the lake, we can use a proportion.
Step 1: Set up the proportion using the tagged fish as a ratio:
tagged fish / total fish = tagged fish in sample / total fish in the lake
(4 / 80) = (70 / x)
Step 2: Cross-multiply and solve for x:
4x = 70 * 80
4x = 5600
Step 3: Divide both sides by 4:
x = 5600 / 4
x = 1400
Therefore, based on the proportion, we estimate that there are 1400 fish in the lake.
To estimate the number of fish in the lake, we can use a technique called mark and recapture. This method assumes that the proportion of tagged fish in the second catch is representative of the proportion of tagged fish in the entire population.
Let's denote the total number of fish in the lake as "N" (which is what we want to estimate), and the number of tagged fish in the first catch as "n". We can set up a proportion based on the assumption mentioned earlier:
n / N = x / M
where:
- n is the number of tagged fish in the first catch (70 in this case),
- N is the total number of fish in the lake (what we're trying to find),
- x is the number of recaptured fish in the second catch (4 tagged fish),
- M is the size of the second catch (80 fish).
Now, let's plug in the known values into the equation:
70 / N = 4 / 80
To solve for N, we cross-multiply and solve for N:
70 * 80 = 4 * N
5600 = 4 * N
N = 5600 / 4
N = 1400
Therefore, based on this estimation, the number of fish in the lake would be approximately 1400.