Problem 73 - 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

The lake has approximately 194444 fish.

Step 1: Set Up Proportion

The key principle in this problem is that the proportion of tagged fish in the sample should represent the proportion of tagged fish in the entire lake. Let \( N \) be the total number of fish in the lake. Then, the proportion of tagged fish in the lake, \( \frac{1000}{N} \), should equal the proportion of tagged fish in the sample, \( \frac{27}{5250} \). This gives the equation: \[ \frac{1000}{N} = \frac{27}{5250} \].

Step 2: Solve for Total Number of Fish

Now, solve the equation \( \frac{1000}{N} = \frac{27}{5250} \) to find \( N \). By cross-multiplying, we get \( 1000 \times 5250 = 27 \times N \).

Step 3: Perform Multiplication and Division

First, calculate \( 1000 \times 5250 = 5250000 \). Then, divide \( 5250000 \) by 27 to solve for \( N \). This gives \( N = \frac{5250000}{27} \).

Step 4: Calculate Final Answer

Perform the division to find \( N \). Thus, \( N = 194444.44 \), which we round to the nearest whole number since there cannot be a fraction of a fish. Therefore, \( N \approx 194444 \).

Key Concepts

Proportion

In the context of the isotope dilution method used in our fish-counting problem, the concept of proportion is fundamental. To understand it simply, a proportion is a statement that two ratios are equal. In this exercise, we use proportion to compare the number of tagged fish to the total fish population, both in a sample and potentially across an entire lake.

Let's break it down:

• The first ratio is the number of tagged fish in your sample to the total number of fish in the sample: \( \frac{27}{5250} \).
• The second ratio is the total number of tagged fish originally released to the total population in the lake: \( \frac{1000}{N} \).

Setting these two ratios equal forms a proportion, which is a powerful tool for solving how many fish are actually in the lake. It's this equality that underpins the method used to calculate the estimated population size—mathematically modeling the problem to find a solution.

Mathematical Modeling

Mathematical modeling plays a crucial role in solving real-world problems, like estimating fish populations. By using mathematics, we can create a model based on assumptions or observed data and apply it to understand and solve a problem.

In this exercise:

• We start by identifying that the tagged fish are evenly dispersed and that the proportions will hold true.
• The model establishes an equation from these proportions: \( \frac{1000}{N} = \frac{27}{5250} \).
• We use algebraic manipulation to solve this equation for \( N \), the unknown total number of fish.

This model allows us to solve for values that are difficult, if not impossible, to measure directly. Mathematical modeling transforms a real-world question into a mathematical problem, offering a structured path to discover an answer.

Tagging and Recapture Technique

The tagging and recapture technique is a vital method in ecological studies for estimating population sizes. This technique often involves marking a sample of individuals, releasing them back into their habitat, and then later recapturing another sample to see how many are marked.

Here's how it works in our exercise:

• We tagged and released 1000 fish into the lake.
• After allowing time for dispersion, we recaptured 5250 fish, of which 27 were tagged.

From this information, we can apply the technique to estimate the total fish population. The method assumes that the proportion of tagged fish in the recaptured sample reflects the proportion in the entire population. It's an indirect yet effective technique, providing insights into population dynamics without the need to count every individual directly.