Question

a botanist collected data on the growth of sunflowers to maturity. she modeled the equation y = x + \frac{1}{2} as the line of best fit. the independent variable, x, is height in feet. what does the slope mean in this context? a the sunflower’s starting height is 3 feet b the sunflower’s starting height is \frac{1}{2} foot c the sunflower’s height increases by an additional 1 foot each week d the sunflower’s height increases by an additional \frac{1}{2} foot each week

Explanation:

Step1: Recall Slope Interpretation

In a linear equation \( y = mx + b \), \( m \) is the slope (rate of change) and \( b \) is the y - intercept (initial value). Here, the equation is \( y = x+\frac{1}{3} \), so slope \( m = 1 \)? Wait, no, wait—wait, maybe the equation is \( y=\frac{1}{3}x + 3 \)? Wait, the user's image: the equation is \( y=x+\frac{1}{3} \)? Wait, no, let's re - examine. Wait, the problem is about a sunflower's height \( y \) (in feet) and weeks \( x \). The line of best fit is \( y = x+\frac{1}{3} \)? Wait, no, maybe the slope is 1? Wait, no, the options: Let's analyze each option.

Option A: "The sunflower’s starting height is 3 feet" — the y - intercept here, if the equation is \( y = mx + b \), if \( b=\frac{1}{3} \), not 3. So A is wrong.

Option B: "The sunflower’s ultimate height is \(\frac{1}{3}\) foot" — the equation is linear, it doesn't have an ultimate height (it would keep increasing), so B is wrong.

Option C: "The sunflower’s height increases by an additional 1 foot each week" — in \( y = mx + b \), \( m \) is the slope. If the equation is \( y=x+\frac{1}{3} \), the slope \( m = 1 \). That means for each increase of 1 in \( x \) (1 week), \( y \) (height) increases by 1 foot. So this matches.

Option D: "The sunflower’s height increases by an additional \(\frac{1}{3}\) foot each week" — the slope here is 1, not \(\frac{1}{3}\), so D is wrong.

Wait, maybe I misread the equation. Wait, maybe the equation is \( y=\frac{1}{3}x + 3 \)? No, the options: Let's check again. The key is the slope interpretation. In a linear model \( y = mx + b \), \( m \) is the rate of change (change in y per change in x). If the equation is \( y=x+\frac{1}{3} \), then \( m = 1 \), so for each week (x increases by 1), y (height) increases by 1 foot. So Option C is correct.

Option D: slope would be \(\frac{1}{3}\) if \( m=\frac{1}{3} \), but here \( m = 1 \), so D is wrong.

So the correct option is C.

Answer:

C. The sunflower’s height increases by an additional 1 foot each week.