For a research project on rodents, 81 groundhogs were tagged and released. Later, researchers counted 490 groundhogs, 18 of which had tags. To the nearest whole number, what is the best estimate for the groundhog population?

This problem can be solved using a proportional reasoning method, commonly referred to as the "mark-recapture" method, to estimate the population of groundhogs.

We are given the following:
- 81 groundhogs were initially tagged and released.
- Later, 490 groundhogs were captured, and 18 of them had tags.

Let \( N \) be the total population of groundhogs, which we are trying to estimate. The proportion of tagged groundhogs in the second sample should approximate the proportion of tagged groundhogs in the entire population:

\[
\frac{81}{N} = \frac{18}{490}
\]

Now, we solve for \( N \) by cross-multiplying:

\[
81 \times 490 = 18 \times N
\]

\[
N = \frac{81 \times 490}{18}
\]

Let’s calculate that:

\[
N = \frac{39690}{18} = 2205
\]

Therefore, the best estimate for the total groundhog population is approximately **2,205**.

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

Here are five related questions:
1. What is the general formula for the mark-recapture method?
2. How does increasing the number of recaptured tagged animals affect the population estimate?
3. What are other real-life scenarios where the mark-recapture method can be applied?
4. What assumptions are made in the mark-recapture method, and how might they affect accuracy?
5. How can variations in sampling size impact the reliability of population estimates?

**Tip**: In any mark-recapture problem, ensuring that the sample size is sufficiently large and representative is crucial for an accurate population estimate.

Estimate the total groundhog population using the mark-recapture method. In this problem, 81 groundhogs were tagged, and later 490 were observed, including 18 tagged ones. The proportional reasoning method helps estimate a population of approximately 2,205. Suitable for Grades 9-11.

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