Question

A Department of Natural Resources (DNR) would like to estimate number of fish in a lake. If he/she catches 142

fish on the first day and tag all of them and release back in the lake. Next day he catches 105 fish and observe
that 15 have tag. Find a solution to the following question based on this data.
a) Find an estimate for the total number of fish in the lake. Be sure to state any assumption, define variables
and build a mathematical model for the problem and then solve.
b) Find the 15 fish with tags what percent of the total number caught in the second day is that. Give your
answer as a whole number in percent.
c) Set up an Excel spreadsheet for this problem to estimate number of fish in the like. Make up two additional
sets of data and enter in the same Excel Spreadsheet and find the number of fish in the lake for each case.

Answer 1

Explanation:

Let's solve each part of the problem step by step:

a) To estimate the total number of fish in the lake, we can use a proportion based on the ratio of tagged fish to the total catch. Let:

- N be the total number of fish in the lake.

- 142 be the number of fish caught on the first day.

- 15 be the number of tagged fish caught on the second day.

- 105 be the total number of fish caught on the second day.

We can set up the following proportion:

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

Plugging in the values:

\[ \frac{15}{105} = \frac{N}{142} \]

Now, cross-multiply and solve for N:

\[ 15 \cdot 142 = 105N \]

\[ N = \frac{15 \cdot 142}{105} \]

\[ N \approx 20.28 \]

So, the estimate for the total number of fish in the lake is approximately 20.28 (rounded to the nearest whole number, it would be 20).

b) To find the percentage of fish with tags in the second day's catch:

\[ \text{Percentage} = \left(\frac{\text{Number of tagged fish}}{\text{Total number caught on the second day}}\right) \times 100 \]

Plugging in the values:

\[ \text{Percentage} = \left(\frac{15}{105}\right) \times 100 = \frac{3}{7} \times 100 \]

Now, calculate the percentage:

\[ \text{Percentage} = \frac{300}{7} \approx 42.86\% \]

So, the 15 fish with tags represent approximately 42.86% of the total number caught on the second day.

c) To set up an Excel spreadsheet for this problem and estimate the number of fish in the lake for two additional cases, you can follow these steps:

1. Create an Excel spreadsheet with the following columns: "Day 1 Catch," "Day 2 Catch," "Tagged Fish," "Total Fish Estimate."

2. In the "Day 1 Catch" column, enter 142 for the first case, and adjust it for the other two cases.

3. In the "Day 2 Catch" column, enter 105 for all cases.

4. In the "Tagged Fish" column, enter 15 for all cases.

5. In the "Total Fish Estimate" column, use the formula mentioned in part (a) to calculate the estimate for each case.

6. Excel will automatically calculate the estimates for each case based on the data you entered.

Now, you have an Excel spreadsheet that estimates the number of fish in the lake for the three cases.