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Suppose a scientist tags 100 fish and releases them into a pond. Later, a fisherman catches 10 fish from the same pond.

1. How many tagged fish did the fisherman catch?
2. What percentage of his sample were tagged?
3. Estimate the number of fish in the pond using the proportion:
[tex]\[
\frac{\text{number of tagged fish}}{\text{total fish caught}} = \frac{100}{x}
\][/tex]

4. Calculate the number of fish in the pond using:
[tex]\[
x = \frac{100 \times 10}{\text{number of tagged fish caught}}
\][/tex]

5. Click on the CALCULATE tab to check your answer.

Repeat the experiment:
- The fisherman releases the fish back into the pond.
- Each day for the next few days, he catches 10 fish, records the number of tagged fish, and releases them.

Record your data for Days 1, 2, 3, and 4.

Sagot :

GinnyAnswer
Sure, let's go through this step-by-step.

### Step-by-Step Solution

1. Initial Setup:
- The scientist tags and releases [tex]\( 100 \)[/tex] fish back into the pond.
- Later, a fisherman catches [tex]\( 10 \)[/tex] fish.

2. Catch Analysis:
- Out of the fish caught, [tex]\( 10 \)[/tex] were found to be tagged.
- The fraction of tagged fish in the sample is thus:
[tex]\[ \frac{\text{number of tagged fish in sample}}{\text{total fish caught}} = \frac{10}{10} = 1 \][/tex]

3. Percentage Calculation:
- To convert this fraction to a percentage:
[tex]\[ \left(\frac{\text{number of tagged fish in sample}}{\text{total fish caught}} \right) \times 100 = 1 \times 100 = 100\% \][/tex]
- Hence, [tex]\( 100\% \)[/tex] of the fish in the sample were tagged.

4. Proportion to Estimate Total Fish:
- Given the high percentage, we can use a proportion to estimate the total number of fish in the pond.
- Let [tex]\( x \)[/tex] be the total number of fish in the pond.
- The proportion is set up as:
[tex]\[ \frac{\text{number of tagged fish}}{\text{total number of fish in the pond}} = \frac{\text{number of tagged fish in the sample}}{\text{total fish caught}} \][/tex]
- Plugging in the known values:
[tex]\[ \frac{100}{x} = \frac{10}{10} \][/tex]
- Simplifying the right side:
[tex]\[ \frac{100}{x} = 1 \][/tex]

5. Solving for [tex]\( x \)[/tex]:
- To solve for [tex]\( x \)[/tex], rearrange the equation:
[tex]\[ x = \frac{100}{1} = 100 \][/tex]

6. Conclusion:
- The estimated total number of fish in the pond is [tex]\( 100 \)[/tex].

### Recording Data for Multiple Days:
- For the subsequent days, the fisherman catches and analyzes [tex]\( 10 \)[/tex] fish each day. The process would be the same:
- Record the number of tagged fish in those samples.
- Calculate the proportion of tagged fish.
- Use the steps above to reassess the estimate each day.

### Example for Further Experimentation:
- If on Day 2, out of 10 fish caught, 5 are tagged:
[tex]\[ \frac{5}{10} = 0.5 \Rightarrow (0.5 \times 100) = 50\% \][/tex]
- Using proportions:
[tex]\[ \frac{100}{x} = \frac{5}{10} \Rightarrow \frac{100}{x} = 0.5 \Rightarrow x = 200 \][/tex]

This continuous data collection and reassessment will refine the estimate over time.

I hope this clear explanation helps you understand the process and how the estimation was made!
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