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 \(a\).

Short answer

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

Step-by-step solution

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.

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.

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 \( c \approx -30.34 \).

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 \( f = 5.26a - 30.34 \).