Question

23 nathan observes the growth of sunflower plants for a science project. he collects data on the relationship between the height, in centimeters, of each sunflower plant during a thirty - day period and the amount of fertilizer, in grams, used on each plant. the equation for the line of best fit for these data is $y = 0.35x + 2$, where $y$ is the height of the plant in centimeters and $x$ is the number of grams of fertilizer used. based on the model, what does the slope of the line represent?
a the height of the plant
b the amount of fertilizer used
c the average growth of the plant per gram of fertilizer used
d the average amount of fertilizer used per centimeter the plant grew

Explanation:

Step1: Recall Slope-Intercept Form

The equation of a line in slope - intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. In the given equation \(y=0.35x + 2\), \(m = 0.35\), \(x\) represents the amount of fertilizer (in grams) and \(y\) represents the height of the plant (in centimeters).

Step2: Interpret the Slope

The slope \(m\) in the equation \(y=mx + b\) represents the rate of change of \(y\) with respect to \(x\). So, the slope of \(0.35\) means that for every 1 - unit increase in \(x\) (which is the amount of fertilizer in grams), \(y\) (the height of the plant in centimeters) increases by \(0.35\) units. In other words, it represents the average growth of the plant (in centimeters) per gram of fertilizer used.

Let's analyze the options:

• Option A: The height of the plant is represented by \(y\), not the slope. So, A is incorrect.
• Option B: The amount of fertilizer used is represented by \(x\), not the slope. So, B is incorrect.
• Option C: As we interpreted, the slope \(0.35\) is the rate of change of height (\(y\)) with respect to fertilizer (\(x\)), which is the average growth of the plant per gram of fertilizer used. So, C is correct.
• Option D: The slope is the change in height per unit change in fertilizer, not the change in fertilizer per unit change in height. If we wanted the change in fertilizer per unit change in height, we would look at the reciprocal of the slope. So, D is incorrect.

Answer:

C. the average growth of the plant per gram of fertilizer used