A biologists wants to approximate the number of fish in a lake. Instead of trying to catch all the fish by dragging nets it is easier (and safer for the fish to avoid the stress) to make small catches at different parts of the lake. When doing this on the first day they caught and tagged 132 fish. Two weeks later they go out to the lake and catch 92 fish and 10 of those are tagged. Estimate the total number of fish in the lake.

This problem involves the **mark and recapture method** for estimating the total population of a species in a given area. The basic idea is to tag a certain number of fish, release them back into the lake, and then later capture a sample of fish to see how many of the tagged fish appear in that sample. This allows us to estimate the total population of fish using proportions.

Let:
- \( N \) be the total number of fish in the lake (which we want to estimate),
- 132 be the number of fish tagged on the first day,
- 92 be the number of fish caught in the second sample,
- 10 be the number of tagged fish in that second sample.

Using the proportion method, we assume that the proportion of tagged fish in the second sample is roughly equal to the proportion of tagged fish in the entire population.

\[
\frac{\text{Number of tagged fish in the second sample}}{\text{Total number of fish in the second sample}} = \frac{\text{Number of tagged fish}}{\text{Total number of fish in the lake}}
\]

Substituting the given values:

\[
\frac{10}{92} = \frac{132}{N}
\]

Now solve for \( N \):

\[
N = \frac{132 \times 92}{10} = 1214.4
\]

Rounding to the nearest whole number, the estimated total number of fish in the lake is approximately **1214**.

Would you like more details or have any questions about this method?

### Related Questions:
1. How does increasing the sample size in the second catch improve the estimate's accuracy?
2. What are the assumptions made by the mark and recapture method?
3. How could environmental factors affect the reliability of the estimate?
4. What would happen to the estimate if no tagged fish were caught in the second sample?
5. How can statistical techniques help refine population estimates from field data?

**Tip**: Repeating the process multiple times and averaging the results can improve the reliability of population estimates.

A biologist wants to approximate the number of fish in a lake. Instead of trying to catch all the fish by dragging nets, it is easier (and safer for the fish to avoid the stress) to make small catches at different parts of the lake. On the first day, they caught and tagged 132 fish. Two weeks later, they caught 92 fish, and 10 of those were tagged. Estimate the total number of fish in the lake.

This problem demonstrates how to estimate the total population of fish in a lake using the Mark and Recapture method. A biologist tagged 132 fish on the first day and later recaptured 92 fish, with 10 being tagged. This method applies proportion calculations to estimate the total fish population to be approximately 1214.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation