Question

the graph shows the relationship between the number of marbles and the volume in a new cylinder. write an equation to represent this line. what does a slope of \\(\frac{1}{2}\\) mean in this situation? the slope is the rate of change, or the amount the volume increases by for every marble added. the volume increases 1 ml for every \\(\frac{1}{2}\\) marble added. the volume is \\(\frac{1}{2}\\) ml when there are no marbles in the cylinder. the volume increases \\(\frac{1}{2}\\) ml for every 1 marble added.

Explanation:

Step1: Recall slope interpretation

Slope $m=\frac{\Delta y}{\Delta x}$, where $\Delta y$ = volume change, $\Delta x$ = marble count change.

Step2: Match slope to context

Slope $\frac{1}{2}$ means $\Delta y=\frac{1}{2}$ when $\Delta x=1$.

Answer:

The volume increases $\frac{1}{2}$ mL for every 1 marble added.

---

Equation for the line:

Step1: Identify y-intercept

From graph, when $x=0$, $y=8$. So $b=8$.

Step2: Write slope-intercept form

Use $y=mx+b$, $m=\frac{1}{2}$, $b=8$.
$$y=\frac{1}{2}x + 8$$