Question

from unit 3, lesson 4
a seed is planted in a glass pot, and its height is measured in centimeters every day.
the best - fit line is given by the equation ( y = 0.404x - 5.18 ), where ( y ) represents the height of the plant above ground level, and ( x ) represents the number of days since it was planted.
a. what is the slope of the best - fit line?
what does the slope of the line mean in this situation?
is it reasonable?
b. what is the ( y ) - intercept of the best - fit line?
what does the ( y ) - intercept of the line mean in this situation?
is it reasonable?

Explanation:

Step1: Identify the slope from the linear equation

The equation of a line is in the form \(y = mx + b\), where \(m\) is the slope. For the equation \(y = 0.404x-5.18\), the slope \(m = 0.404\).
In the context of the problem, \(x\) is the number of days and \(y\) is the height. The slope represents the rate of change. So, it means the plant grows approximately \(0.404\) centimeters per day. This is reasonable as plants do grow over time, and a growth rate of about \(0.4\) cm per day is a plausible value for a young plant.

Step2: Identify the y - intercept from the linear equation

In the equation \(y=mx + b\), \(b\) is the y - intercept. For \(y = 0.404x-5.18\), the y - intercept \(b=-5.18\).
When \(x = 0\) (the day the seed is planted), \(y=-5.18\). This would mean the height of the plant is \(- 5.18\) cm at \(x = 0\). Since height cannot be negative in a real - world sense (assuming we are measuring height above the ground level), this is not reasonable. It might be due to the limitations of the linear model (e.g., extrapolating too far from the data points near \(x = 0\)).

Answer:

a.

• The slope of the best - fit line is \(0.404\).
• The slope means the plant grows approximately \(0.404\) centimeters per day.
• It is reasonable as it represents a positive growth rate which is expected for a plant.

b.

• The \(y\) - intercept of the best - fit line is \(-5.18\).
• The \(y\) - intercept means that at \(x = 0\) (the day the seed is planted), the height of the plant is \(-5.18\) cm.
• It is not reasonable because height (above ground level) cannot be negative.