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: Identify line equation form

We use slope-intercept form $y = mx + b$, where $m$ is slope, $b$ is y-intercept.

Step2: Find y-intercept $b$

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

Step3: Confirm slope meaning

Slope $m=\frac{1}{2}$ means volume ($y$) increases by $\frac{1}{2}$ per marble ($x$).

Step4: Write the line equation

Substitute $m=\frac{1}{2}$ and $b=8$ into the form:
$y = \frac{1}{2}x + 8$

Answer:

Equation: $\boldsymbol{y = \frac{1}{2}x + 8}$
Correct slope interpretation: The volume increases $\frac{1}{2}$ mL for every 1 marble added.