Problem 54 - Algebra 1

Question

At nine days old, a flight feather of a night heron is 17 millimeters. At 30 days, the flight feather is 150 millimeters. Write a linear model that gives feather length \(f\) in terms of age \(a\).

Step-by-Step Solution

Answer

The linear model that gives feather length \(f\) in terms of age \(a\) is \( f = 5.26a - 30.34 \)

Step 1: Identify the given data points

The problem presents two data points which are (9,17) and (30, 150) where the first element in the pair is the age \(a\) in days and the second is the feather length \(f\) in millimeters.

Step 2: Calculate the slope of the line

The general formula for calculating the slope (m) of a line given two points (x1, y1), (x2, y2) is \[ m = \frac{y2 - y1}{x2 - x1} \]. Using our points (9, 17) and (30, 150), the slope of our line would be \[ m = \frac{150 - 17}{30 - 9} \] which evaluates to approximately 5.26.

Step 3: Find the y-intercept of the line

Next, the general form of a linear equation is \(y = mx + c\) where c is the y-intercept. We can find c by substituting one of our points and the calculated slope into this equation and solve for c. Using point (9, 17) and slope 5.26, we get \[ 17 = 5.26 * 9 + c \]. Solving this gives \( c \approx -30.34 \).

Step 4: Write the Linear Model

As the final step, insert the calculated slope and y-intercept into the general form of a linear equation \( y = mx + c \). This will give us our feather growth model in relation to age \( f = 5.26a - 30.34 \).

Key Concepts

Slope Calculation

The slope of a line is a measure of its steepness, represented mathematically by the letter \( m \). It's crucial for finding how one variable changes in relation to another in a linear equation.
The formula for slope is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct data points on the line.
In this heron feather problem, we use the points \((9, 17)\) and \((30, 150)\).

• For these points, substitute into the formula: \(m = \frac{150 - 17}{30 - 9} \).
• This results in a slope \( m \approx 5.26 \), indicating the feather grows approximately 5.26 millimeters per day.

Understanding the slope helps us see how quickly the feather length increases with age.

Data Points

Data points are specific values plotted on a graph which can represent real-world scenarios.
In linear equations, they show how two quantities are related.
In our example, the data points are \((9, 17)\) and \((30, 150)\), showing that a feather is 17 mm long at 9 days and 150 mm at 30 days.

• The first number in each pair is age in days, \(a\).
• The second number is the feather length in millimeters, \(f\).

Using these points allows us to derive a model to predict feather growth.

Y-Intercept

The y-intercept, represented by \( c \), is where the line crosses the y-axis.
It shows the value of the dependent variable when the independent variable is zero.
In a linear equation like \( y = mx + c \), \( c \) provides a starting point for the trend.
To find the y-intercept for our problem:

• Use one of the data points, say \((9, 17)\), and the slope \(5.26\).
• Substitute into the equation: \( 17 = 5.26 \times 9 + c \).
• Solving this gives \( c \approx -30.34 \).

This means at age zero, the feather length is roughly \(-30.34\) millimeters, indicating our equation models a theoretical extension backward in time.

Linear Model

Creating a linear model helps describe relationships between variables in a straightforward way.
It uses the slope and y-intercept to express one variable as a function of another.
In this example, the linear model is \( f = 5.26a - 30.34 \).

• The slope (5.26) shows the rate of feather growth per day.
• The y-intercept (-30.34) suggests an initial theoretical length if the model were extended to age zero.

Such models allow predictions, meaning we can estimate feather length at any given age, which is helpful for understanding growth patterns.