Problem 51 - Prealgebra

Question

The Google Earth map shows Yellowstone National Park. There is an average of 2.3 moose per square mile. If there are about \(7,986\) moose in Yellowstone, how many square miles does Yellowstone cover? Round to the nearest square mile. (THE IMAGES CANNOT COPY)

Step-by-Step Solution

Answer

Yellowstone covers approximately 3,472 square miles.

Step 1: Understand the Problem

We know that there are on average 2.3 moose per square mile in Yellowstone National Park. The total number of moose is about 7,986. We need to find out the total area in square miles where these moose are distributed.

Step 2: Set Up the Equation

Let \(x\) represent the total number of square miles in Yellowstone. Since there are on average 2.3 moose per square mile, we can write the equation: \(2.3x = 7,986\).

Step 3: Solve for the Unknown

To find \(x\), divide both sides of the equation by 2.3. This gives \(x = \frac{7,986}{2.3}\).

Step 4: Divide and Simplify

Calculate \(x\) by performing the division: \(x = 3,472.17\).

Step 5: Round the Answer

Since we need to round to the nearest square mile, we round 3,472.17 to 3,472.

Key Concepts

Average Calculations

The concept of average helps us understand how a total amount is distributed evenly over a set. For example, in this exercise, the average offers the information that there are 2.3 moose per square mile in Yellowstone National Park. When you know an average and the total count of items in question, you can find out how many sets there are. This means if you know the total number of moose and the average moose per square mile, you can determine how many square miles contain moose.

To calculate this, set up an equation where the average equals the total divided by the count of sets. If the average of moose equals the total number of moose divided by the square miles, the relation is simple. This makes it easier to manage large amounts of data without tracking each individual entry. Using averages can simplify problem-solving in real-life situations.

Division

Division is a fundamental arithmetic operation used to determine how many times one number "fits" into another. In relation to average calculations, division helps determine the number of "sets" or groups based on a total and its average. For the Yellowstone moose problem, the average of 2.3 moose per square mile was used to set up the division problem.

To find the number of square miles, divide the total number of moose (7,986) by the average number of moose per mile. The equation is set as \(2.3x = 7,986\). Solving for \(x\), we divide both sides by 2.3 resulting in \(x = \frac{7,986}{2.3}\).
This operation gives us a precise figure of how the moose are spread across the park. Division takes the complex totalities and breaks them into understandable parts, making it easier for analysis and decision-making.

Rounding Numbers

Rounding numbers is a strategy used to simplify numbers by reducing the digits while keeping the value close to the original. This is especially useful when dealing with decimal points or when precision beyond a certain point is unnecessary. In problems like calculating square miles for moose distribution in a vast area, too much precision can be impractical, thus rounding becomes important.

In our Yellowstone National Park problem, the division resulted in \(3,472.17\) square miles, which isn't a practical figure for reporting territory size. We round to the nearest whole number, which involves checking the digit after the decimal. If it's 5 or more, we round up. If less, round down. Here, 0.17 suggests rounding down, making the simpler answer 3,472 square miles.
Thus, rounding takes lengthy decimal expressions and offers a straightforward solution, aiding in clearer communication and understanding in mathematics as well as everyday scenarios.