Problem 50 - College Algebra with Corequisite Support

Question

In \(2003,\) the owl population in a park was measured to be \(340 .\) By 2007 the population was measured again to be 285 . The population changes linearly. Let the input be years since \(2003 .\) (a) Find a formula for the owl population, \(P\). Let the input be years since \(2003 .\) (b) What does your model predict the owl population to be in \(2012 ?\)

Step-by-Step Solution

Answer

The formula for the population is \(P(t) = -13.75t + 340\) and in 2012, the population is approximately 216.

Step 1: Define Variables

We need to define our variables first. Let the number of years since 2003 be represented by the variable \(t\). The owl population at time \(t\) can be denoted by \(P(t)\). In 2003, which is \(t = 0\), the population \(P(0) = 340\). In 2007, which is \(t = 4\), the population \(P(4) = 285\).

Step 2: Calculate the Rate of Change

The change in population between 2003 and 2007 is \(285 - 340 = -55\). This change occurs over a period of \(4\) years. The rate of change, or slope \(m\), is \( \frac{-55}{4} = -13.75\).

Step 3: Write the Equation of the Line

Since population changes linearly, use the point-slope form of the line equation: \(P(t) = mt + b\), where \(m\) is the slope and \(b\) is the y-intercept. From Step 1, the y-intercept is the population in 2003, hence \(b = 340\). The equation becomes \(P(t) = -13.75t + 340\).

Step 4: Calculate Population in 2012

To find the population in 2012, calculate \(t\) for 2012. Since 2012 is \(9\) years after 2003, \(t = 9\). Substitute \(t = 9\) into the equation: \(P(9) = -13.75 \times 9 + 340\).

Step 5: Solve for P(9)

Compute \(-13.75 \times 9 = -123.75\). Thus, \(P(9) = -123.75 + 340 = 216.25\).

Key Concepts

Rate of Change

In any linear relationship, understanding the concept of rate of change is essential. The rate of change in a linear equation is often referred to as the "slope" or the "gradient". This measures how much one quantity changes with respect to a change in another quantity. In the context of our owl population problem, it helps us understand how the population decreases over time.

In this specific scenario, the rate of change tells us how many owls are lost each year. With the given data from 2003 (population = 340 ows) and 2007 (population = 285 owls), we can determine how quickly the population is declining. Calculate this using the formula for slope:

• Change in population = Final population - Initial population = 285 - 340 = -55 owls
• Change in years = 2007 - 2003 = 4 years
• Rate of change (slope, m) = \( \frac{-55}{4} \) = -13.75 owls per year

A rate of -13.75 means that, on average, the owl population decreases by 13.75 owls every year.

Linear Equation

Linear equations form the foundation of understanding how quantities change consistently over time. A linear equation is characterized by two main components: a constant rate of change and an initial value.

In our exercise, the population change of the owls is a linear phenomenon, which can be expressed by the linear equation \( P(t) = mt + b \). Here,

• \( m \) is the rate of change, or slope, which in this case is \( -13.75 \).
• \( b \) is the y-intercept, representing the initial population in the year 2003, which is 340 owls.
• \( t \) is the number of years since 2003.

Thus, the linear equation for the owl population is formulated as \( P(t) = -13.75t + 340 \). This equation models the population decline over time, predicting the number of owls left after certain years. It is a straightforward way to visualize and compute changes happening at a constant rate.

Point-Slope Form

The point-slope form of a line is an effective tool to derive linear equations when you know one point on the line and the rate of change. It is expressed as \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a known point on the line, and \( m \) is the slope.

In the context of the owl population problem, we started with a known point:

• In 2003 (when \( t = 0 \)), the population was 340, giving us the point \( (0, 340) \).

The point-slope formula rearranges into the slope-intercept form by substituting:

• The known point \( (t_1, P(t_1)) = (0, 340) \)
• And the slope \( m = -13.75 \)

The equation becomes \( P(t) = -13.75(t - 0) + 340 \), simplifying to \( P(t) = -13.75t + 340 \).
This form is valuable because it builds on easily obtainable information and helps us quickly formulate and understand the linear model of a given scenario.