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!
