To determine the number of trouttrout in a lakelake​, a conservationistconservationist catches 265265 trouttrout​, tags​ them, and releases them. Later 137137 trouttrout are​ caught, and it is found that 6666 of them are tagged. Estimate how many there are in the lakelake.
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Part 1
The approximate number of trouttrout is enter your response here. 
​(Round to the nearest whole number as​ needed.)

This problem involves using a proportion to estimate the number of trout in the lake, based on the concept of **capture-recapture**.

Here's how we solve it step by step:

1. **Define variables:**
   - Let \( N \) represent the total number of trout in the lake (the value we need to estimate).
   - The conservationist tags and releases 265 trout.
   - In a later sample of 137 trout, 66 of them are tagged.

2. **Set up the proportion:**
   - The proportion of tagged trout in the lake should be approximately equal to the proportion of tagged trout in the sample.
   - So, we set up the equation:
   \[
   \frac{\text{Tagged Trout (first capture)}}{\text{Total Trout in Lake}} = \frac{\text{Tagged Trout in Sample}}{\text{Total Trout in Sample}}
   \]
   - Using the given numbers, the equation becomes:
   \[
   \frac{265}{N} = \frac{66}{137}
   \]

3. **Solve for \( N \):**
   - Cross-multiply to solve for \( N \):
   \[
   265 \times 137 = 66 \times N
   \]
   \[
   36,305 = 66N
   \]
   \[
   N = \frac{36,305}{66} \approx 550.08
   \]

4. **Round the answer:**
   - Rounding \( N \) to the nearest whole number gives approximately:
   \[
   N \approx 550
   \]

Thus, the estimated total number of trout in the lake is **550**.

Would you like more details or have any questions? 

Here are 5 related questions to expand your understanding:
1. How is the capture-recapture method used in wildlife population studies?
2. What assumptions must hold true for the capture-recapture method to be accurate?
3. How would the estimate change if the second sample had fewer tagged trout?
4. Can this method be used for estimating populations of species other than fish?
5. What are some potential sources of error in a capture-recapture study?

**Tip:** Always ensure the tagged population has mixed thoroughly with the rest of the population before recapturing. This ensures accurate results!

To determine the number of trout in a lake, a conservationist catches 265 trout, tags them, and releases them. Later 137 trout are caught, and it is found that 66 of them are tagged. Estimate how many there are in the lake. (Round to the nearest whole number as needed.)

Learn how to estimate the number of trout in a lake using the capture-recapture method. This example involves tagging 265 trout and using proportional reasoning to calculate the total population after catching a sample of 137 trout, 66 of which were tagged. Suitable for students in Grades 9-11.

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