Proportions in Bear Population Estimation

Categories: Biology

Essay, Pages 3 (502 words)

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Proportions exist in many real-world applications, including the estimation of the size of the bear population on the Keweenaw Peninsula. By comparing data from two experiments, conservationists are able to predict patterns of animal increase or decrease.

In this scenario, 50 bears were captured, tagged, and released to estimate the size of the bear population. A year later, after capturing a random sample of 100 bears, only 2 of the bears captured were tagged bears. These proportions will be used to determine the bear population on the peninsula.

This new bear scenario can be solved by applying the concept of proportions, which allows us to assume that the ratio of originally tagged bears to the whole population is equal to the ratio of recaptured tagged bears to the size of the sample.

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To determine the estimated solution, the bears will be the extraneous variables that will be defined for solving the proportions used.

Proportions Setup

The ratio of originally tagged bears to the whole population: _{50 / X}

The ratio of recaptured tagged bears to the sample size 100: _{2 / 100}

This is the proportion set up and ready to solve:

_{50 / X} = _{2 / 100}

The next step is to cross multiply:

5000 = 2X

Divide both sides by 2:

X = 2500

The bear population on the Keweenaw Peninsula is estimated to be around 2500.

Linear Equation for Y

For the second problem in this assignment, the equation must be solved for Y.

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Continuing the discussion of proportions, a single fraction (ratio) exists on both sides of the equal sign, so it can be solved by cross-multiplying the extremes and means:

Y - 1 / X + 3 = -3 / 4

4(Y - 1) = -3(X + 3)

4Y - 4 = -3X - 9

Add 4 to both sides:

4Y = -3X - 5

Divide both sides by 4:

Y = -3X - 5

This is a linear equation in the form of y = mx + b. After comparing the solution to the original problem, it is noticed that the slope, -3/4, is the same number on the right side of the equation. This indicates that another method exists for solving the same equation:

Y - 1 / X + 3 = -3 / 4

4(Y - 1) = -3(X + 3)

Y - 1 = -3(X + 3) / 4

Add 1 to both sides:

Y = -3(X + 3) / 4 + 1

Y = -3X / 4 - 9 / 4 + 1

Y = -3X / 4 - 5 / 4

After solving both of these problems, it's interesting to see how two different equations can be solved using the same basic mathematical functions. It also illustrates how everyday life can incorporate these mathematical functions to solve or estimate various daily life events for different reasons.

Conclusion

In conclusion, the concept of proportions proved to be a valuable tool in estimating the bear population on the Keweenaw Peninsula. By using the data from tagged bears and their recapture rates, we were able to estimate a population size of approximately 2500 bears.

Furthermore, we demonstrated how linear equations can be utilized to solve problems with the same mathematical functions, highlighting the versatility of mathematical concepts in addressing various real-world scenarios. This serves as a reminder of how mathematics can play a crucial role in solving and estimating everyday life events for a range of purposes.