Problem 61 - Chemistry and Chemical Reactivity

Question

The principle underlying the isotope dilution method of analysis can be applied to many kinds of problems. Suppose that you, a marine biologist, want to estimate the number of fish in a lake. You release 1000 tagged fish, and after allowing an adequate amount of time for the fish to disperse evenly in the lake, you catch 5250 fish and find that 27 of them have tags. How many fish are in the lake?

Step-by-Step Solution

Answer

Approximately 194,444 fish are in the lake.

Step 1: Understand the Problem

We are tasked with estimating the total number of fish in a lake. We know that 1000 tagged fish were released, and after some time, a sample of 5250 fish was caught, which included 27 tagged fish. We need to determine the total fish population based on this sample.

Step 2: Set Up the Proportion

The proportion of tagged fish in the sample caught should be equivalent to the proportion of tagged fish in the entire lake. Let the total number of fish in the lake be represented by \( N \). The equation for the proportion is \( \frac{27}{5250} = \frac{1000}{N} \).

Step 3: Solve the Proportion Equation

We need to solve the equation \( \frac{27}{5250} = \frac{1000}{N} \) to find \( N \). This can be done by cross multiplying to get: \( 27N = 5250 \times 1000 \).

Step 4: Calculate the Total Number of Fish

Solve for \( N \) by performing the multiplication on the right side of the equation and then divide by 27: \( N = \frac{5250 \times 1000}{27} \). This simplifies to \( N \approx 194,444.44 \). Since the number of fish must be an integer, \( N \approx 194,444 \).

Step 5: Interpret the Result

The calculated number, \( N \), represents the estimated total number of fish in the lake. Given that this is an estimate based on sampling, it is understood to be an approximation, but it suggests there are approximately 194,444 fish in the lake.

Key Concepts

Population Estimation

Estimating a fish population in a lake is a common problem in marine biology and ecological studies. By releasing a known number of tagged fish into a lake and later recapturing a sample, we can estimate the total population. This approach is used to understand the size of the population without having to catch every single fish, which would be impractical.

The key idea is that the proportion of tagged fish in the sample should be similar to the proportion of tagged fish in the entire population. By establishing this proportion, we can solve for the total population. This method provides an approximation and relies on assumptions such as initial tagged fish mixing evenly with the rest of the population.

Proportion Calculation

Calculating proportions is central to the problem of estimating the fish population. Our task is to set up an equation where the proportion of tagged fish in a sample is equal to their proportion in the entire lake population.

In this exercise, we know 1000 fish were tagged, and within a sample of 5250, there were 27 tagged fish. Using these numbers, we set up the equation:

• Proportion in sample: \[ \frac{27}{5250} \]
• Proportion in lake: \[ \frac{1000}{N} \] where \( N \) is the total number of fish.

By solving the equation \( \frac{27}{5250} = \frac{1000}{N} \), we determine \( N \), which gives an estimate of total fish. Understanding proportion calculation is crucial, as it allows us to break down the problem into manageable steps and relate parts of a whole to one another.

Tagged Fish Analysis

Analyzing the tagged fish after they are released gives insights into the overall fish population. The concept relies on the idea that the tagged fish, once mixed completely with the non-tagged population, act as a representation of the total fish in the lake. This is crucial because the initial setup must ensure enough time for dispersement, so the tagged and non-tagged fish mingle fully.

In this exercise, we tagged 1000 fish, and after recapturing a sample, only 27 tagged fish were present in the catch. This forms the basis of our proportion calculation. Thus, analyzing how many tagged fish are recaptured enables biologists to draw conclusions about the broader population without having to rely on more invasive or damaging practices which could harm the ecosystem.

Sampling Method in Biology

The sampling method used in biology, such as in this fish population problem, involves taking a manageable subset of a larger group to make inferences about the entire group. This approach is important for practical and ethical reasons, as it reduces the need for total capture, which can disrupt the environment and stress wildlife.

In our example, the sample size was 5250 fish, which is a balance between being large enough to provide a reliable estimate and small enough to be practical. Proper sampling involves ensuring the sample is random and that the population is well-mixed, which helps ensure that the proportion of tagged to untagged fish in the sample reflects the actual ratio in the entire lake.

This method helps students and researchers alike understand ecological dynamics and population biology, making it an invaluable tool in field studies and conservation efforts.