Problem 52 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Estimating Seal Pups in a Breeding Area According to an actual survey in \(1961,\) to estimate the number of seal pups in a certain breeding area in Alaska, 4963 pups were tagged in early August. In late August, a sample of 900 pups was examined and 218 of these were found to have been tagged. Use this information to estimate, to the nearest hundred, the total number of seal pups in this breeding area.

Step-by-Step Solution

Answer

Approximately 20400 seal pups.

Step 1: Understand the Use of Proportions

Proportions can be used to estimate the total number of seal pups by comparing the tagged pups in the examined sample to the total tagged population. The proportion of tagged pups in the sample should match the proportion of tagged pups in the entire breeding area.

Step 2: Establish a Proportion Equation

Let \( N \) be the estimated total number of seal pups in the breeding area. Using the proportion of tagged pups in the sample (218 tagged out of 900 total), set up the equation: \( \frac{218}{900} = \frac{4963}{N} \).

Step 3: Solve the Proportion Equation

Cross-multiply to solve for \( N \). This gives us \( 218N = 900 \times 4963 \). Simplifying, we have \( 218N = 4466700 \).

Step 4: Perform the Calculation

Divide both sides of the equation by 218 to find \( N \). Therefore, \( N = \frac{4466700}{218} \approx 20485.77 \).

Step 5: Round the Result

Round the estimated total number of seal pups to the nearest hundred. Thus, \( N \approx 20400 \).

Key Concepts

Population Estimation

Population estimation is crucial in understanding wildlife numbers, distribution, and the dynamics of various species. In the context of our exercise, it involves predicting the total number of seal pups based on a sample survey. This method relies on the idea that a small, manageable subset of the population (the sample) can be used to make educated guesses about the larger group. Without measuring each individual in a population, researchers can still obtain a reliable estimate if a significant sample size is considered.

To estimate populations effectively, scientists often employ sample surveys combined with tag mark-recapture methods. Here's how it generally works:

• Mark a known number of individuals from the population (in this case, the seal pups tagged in early August).
• Later, survey a subset of the population and note how many of them are marked (sample examined in late August).

From these numbers, you can calculate the total population size using proportions, which we'll explore further. This approach minimizes time and resources while maximizing data efficiency.

Cross Multiplication

Cross multiplication is a mathematical method used to solve equations involving two fractions set equal to each other, known as proportions. It's a handy tool in the estimation of populations, such as seal pups in our example.

Let's break down the process:

• Given a proportion equation, such as \( \frac{218}{900} = \frac{4963}{N} \), it lets us determine the unknown variable (in this case, \( N \), the total population).
• To solve for \( N \), you cross multiply, creating an equation without fractions: \( 218N = 900 \times 4963 \).

This method simplifies finding unknowns by allowing us to set the products of the means and extremes equal and solve through basic algebraic steps. By mastering cross multiplication, you'll find it easier to deal with proportion equations, especially in biology and other applied sciences.

Proportion Equation

A proportion equation is a statement that two ratios are equivalent. In our seal pup estimation, it expresses the idea that the tagged portion of the sample should statistically resemble the tagged portion of the whole population.

This is how it was set up in the exercise:

• The ratio of tagged pups in the sample, \( \frac{218}{900} \), is set to be equivalent to the tagged ratio in the total population, \( \frac{4963}{N} \).
• The proportion equation thus formed, \( \frac{218}{900} = \frac{4963}{N} \), links the known sample data with the unknown total population \( N \).

By using such equations, you can seamlessly transition from real-world sampling data to estimating entire populations. The accuracy of your estimation greatly depends on how representative and random the sample is compared to the whole population.

Wildlife Management

Wildlife management is the practical application of ecological knowledge to wildlife populations through scientific methods to balance the needs of animals and people. Estimating populations like seal pups is a valuable skill within wildlife management because it helps track and manage animal populations effectively.

Imagine a wildlife manager:

• They use the estimated data to make decisions about conservation strategies and habitat protection.
• Through understanding population trends, they can respond to emerging issues, like species at risk due to environmental changes or human activities.

By regularly assessing and managing wildlife populations, managers can ensure sustainable ecosystems. They use population data to determine if species are increasing, decreasing, or stable, thus informing conservation efforts and policy-making decisions.