Problem 89 - Algebra for College Students

Question

Solve each problem. Capture-recapture method. To estimate the size of the grizzly bear population in a national park, rangers tagged and released 12 bears. Later it was observed that in 23 sightings of grizzly bears, only two had been tagged. Assuming the proportion of tagged bears in the later sightings is the same as the proportion of tagged bears in the population, estimate the number of bears in the population.

Step-by-Step Solution

Answer

The estimated number of bears is 138.

Step 1 - Define Variables

Let N represent the total bear population in the park, and let T represent the number of tagged bears. We know from the problem that T = 12 (tagged bears).

Step 2 - Set Up the Proportion

According to the problem, the proportion of tagged bears in the initial tagging equals the proportion of tagged bears in the later sightings. This can be set up as \ \( \frac{T}{N} = \frac{2}{23} \).

Step 3 - Substitute Known Values

Substitute \( T = 12 \) into the proportion equation: \ \( \frac{12}{N} = \frac{2}{23} \).

Step 4 - Solve for N

To solve for N, cross-multiply to get: \ \( 12 \cdot 23 = 2 \cdot N \). This simplifies to: \ \( 276 = 2N \), and then divide both sides by 2: \ \( N = 138 \).

Step 5 - Interpret the Result

After solving the equation, we find that the estimated total bear population in the park is 138.

Key Concepts

Proportion

The concept of a proportion is essential to understand the capture-recapture method.

A proportion is an equation that states that two ratios are equal. In this problem, we use proportions to compare the ratio of tagged bears to the total bear population and the ratio of tagged bears observed to the total number of bears spotted. By setting these ratios equal to each other, we can solve for the unknown total population.

For example, if we observe that 2 out of 23 sightings are tagged bears, we can set up the proportion \(\frac{2}{23}\) to represent the tagged sightings and \(\frac{12}{N}\) to represent the entire bear population, where 12 is the number of tagged bears we originally released.

Estimation

Estimation is a crucial mathematical skill, especially when it comes to dealing with large populations that are hard to count directly.

In this exercise, we estimate the total population of bears using the information gathered from a subset of that population. By making an educated guess based on partial data, we can save time and resources. Estimation allows us to make informed decisions without requiring precision that may be impractical or impossible to achieve.

This method relies on the assumption that the small sample we observed is representative of the larger population, which is why the accuracy of our sightings and tags is critical.

Wildlife Population Estimation

Wildlife population estimation is a method used by ecologists to estimate the number of animals in a given area.

It typically involves capturing a sample of animals, tagging or marking them, and then releasing them back into the wild. After some time, a second sample is captured, and the number of tagged animals in this second sample is recorded.

Based on the proportion of tagged to untagged animals in the second sample, scientists can estimate the total population using the formula: \(\frac{T}{N} = \frac{\text{Tagged in Sample}}{\text{Total in Sample}}\). This formula allows us to calculate an estimation of the total population (N) given the number of tagged animals initially released (T) and the observations in the second sample.

Algebraic Equations

Algebraic equations play a crucial role in solving this problem. To estimate the total bear population, we set up an algebraic equation using proportions.

Let's break down the steps:

• First, define the variables. Let N be the total bear population and T be the tagged bears.
• We know T = 12, because the rangers tagged and released 12 bears.
• Next, we use the proportion \(\frac{T}{N}\) to represent the ratio of tagged bears to the whole population.
• Replace T with 12:
• \( \frac{12}{N} = \frac{2}{23} \)
• Now, solve for N by cross-multiplying:
• \(12 \times 23 = 2 \times N\)
• This simplifies to 276 = 2N, then divide both sides by 2:
• N = 138.

So, we estimate there are about 138 bears in the park.