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Chapter 2: Problem 282

For the following exercises, consider this scenario: In 2000, the moose population in a park was measured to be 6,500. By 2010, the population was measured to be 12,500. Assume the population continues to change linearly. Find a formula for the moose population, \(P\).

Short Answer

Expert verified
The formula for the moose population is \(P = 600t - 1193500\).

Step by step solution

01

Identify the Known Values

We are given two data points. In the year 2000, the moose population was 6500, so our first point is (2000, 6500). In 2010, the population was 12,500, giving us a second point, (2010, 12500). These points represent the time in years and the population size at those times.
02

Determine the Slope of the Line

To find a linear equation, we first need the slope, which represents the rate of change in the population. The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the known values: \(m = \frac{12500 - 6500}{2010 - 2000} = \frac{6000}{10} = 600\). The slope is 600.
03

Write the Equation of the Line

Now that we have the slope, we can use the point-slope form to write the equation of the line. The point-slope form is \(y - y_1 = m(x - x_1)\). We'll use the point (2000, 6500): \(P - 6500 = 600(t - 2000)\), where \(P\) is the population and \(t\) is the year.
04

Simplify the Equation to Slope-Intercept Form

Convert the equation to the slope-intercept form \(P = mt + b\) to make it easier to use. Simplify the expression: \(P - 6500 = 600(t - 2000)\) becomes \(P - 6500 = 600t - 1200000\). Add 6500 to both sides: \(P = 600t - 1193500\). Thus, the formula for the population is \(P = 600t - 1193500\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Calculation
The concept of slope is fundamental in understanding linear equations. It tells us how steep a line is, and in real-world situations, it represents a rate of change. Imagine tracking how the moose population changes over time. To find the slope, you use the formula:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Here, \( m \) stands for the slope, and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two points on the line.

For our example with moose population, the points are (2000, 6500) and (2010, 12500). Substituting these values, we calculate:
  • \( m = \frac{12500 - 6500}{2010 - 2000} = \frac{6000}{10} = 600 \)
Thus, the slope of 600 indicates that every year, the moose population increases by 600 individuals. This kind of information helps us predict future changes.
Point-Slope Form
Once you understand the slope, the next step is to use it in the point-slope form of a linear equation. This form is particularly helpful when you know a point on the line and its slope. The point-slope form is written as:
  • \( y - y_1 = m(x - x_1) \)
This formula lets us create an equation for the line. The variables \( (x_1, y_1) \) are coordinates of a known point, and \( m \) is the line's slope.

Let's apply this to our moose population example using the point (2000, 6500) and the slope 600:
  • \( P - 6500 = 600(t - 2000) \)
This equation shows how we can calculate the population \( P \) any year \( t \) by starting from the known point in 2000.
Slope-Intercept Form
Finally, converting to slope-intercept form makes the linear equation even easier to read and use. This form is expressed as:
  • \( y = mx + b \)
Where \( m \) is the slope and \( b \) is the y-intercept 鈥� the point where the line crosses the y-axis.

For the moose population equation, we transform the point-slope equation into slope-intercept form:
  • Start with \( P - 6500 = 600(t - 2000) \)
  • Then simplify: \( P - 6500 = 600t - 1200000 \)
  • Add 6500 to both sides to find \( b \): \( P = 600t - 1193500 \)
This equation clearly shows the population at any point \( t \), making it a powerful tool for predicting future population sizes. In this context, the intercept \(-1193500\) represents a hypothetical starting point backward in time when the population would be zero.

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