Questions: To estimate the number of fish in a lake, scientists use a tagging and recapturing technique. A number of fish are captured, tagged, and then released back into the lake. After a while, some fish are captured and the number of tagged fish is counted. Let T be the total number of fish captured, tagged, and released into the lake, n the number of fish in a recaptured sample, and t the number of fish found tagged in the sample. Finally, let x be the number of fish in the lake. The assumption is that the ratio between tagged fish and the total number of fish in any sample is approximately the same, and hence scientists assume t/n = T/x. Suppose 151 fish were captured, tagged, and released. Then 22 fish were recaptured, and among them 11 were found to be tagged. Estimate the number of fish in the lake. There are approximately fish in the lake.

To estimate the number of fish in a lake, scientists use a tagging and recapturing technique. A number of fish are captured, tagged, and then released back into the lake. After a while, some fish are captured and the number of tagged fish is counted.

Let T be the total number of fish captured, tagged, and released into the lake, n the number of fish in a recaptured sample, and t the number of fish found tagged in the sample. Finally, let x be the number of fish in the lake. The assumption is that the ratio between tagged fish and the total number of fish in any sample is approximately the same, and hence scientists assume t/n = T/x. Suppose 151 fish were captured, tagged, and released. Then 22 fish were recaptured, and among them 11 were found to be tagged. Estimate the number of fish in the lake.

There are approximately fish in the lake.

Transcript text: To estimate the number of fish in a lake, scientists use a tagging and recapturing technique. A number of fish are captured, tagged, and then released back into the lake. After a while, some fish are captured and the number of tagged fish is counted. Let $T$ be the total number of fish captured, tagged, and released into the lake, $n$ the number of fish in a recaptured sample, and t the number of fish found tagged in the sample. Finally, let x be the number of fish in the lake. The assumption is that the ratio between tagged fish and the total number of fish in any sample is approximately the same, and hence scientists assume $\frac{t}{n}=\frac{T}{x}$. Suppose 151 fish were captured, tagged, and released. Then 22 fish were recaptured, and among them 11 were found to be tagged. Estimate the number of fish in the lake. There are approximately $\square$ fish in the lake.

Solution

Solution Steps

Step 1: Identify the given values

Total number of fish tagged and released, $ T = 151 $.
Number of fish recaptured, $ n = 22 $.
Number of tagged fish in the recaptured sample, $ t = 11 $.

Step 2: Use the proportion formula

The proportion formula is given by: \[ \frac{t}{n} = \frac{T}{x} \] where $ x $ is the total number of fish in the lake.

Step 3: Solve for $ x $

Substitute the known values into the formula: \[ \frac{11}{22} = \frac{151}{x} \] Simplify the left side: \[ \frac{1}{2} = \frac{151}{x} \] Cross-multiply to solve for $ x $: \[ x = 2 \times 151 \] \[ x = 302 \]

Final Answer

There are approximately $\boxed{302}$ fish in the lake.

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