Question

ms. joseph is dropping marbles into a cylinder one at a time. she records the volume of what’s in the cylinder after each marble is dropped in. we know that the initial volume in the cylinder was 10 milliliters, which we already have in our equation. now we need to add the total volume of the marbles being added to the cylinder. you can think of that as the volume per marble, 2 ml, that’s the slope, times the number of marbles in the cylinder, which we can represent with the variable x. start by showing that the volume per marble is 2. slope: 2 vertical intercept: 10 volume = initial volume + volume of marbles v = 10 + volume per marble in ml · number of marbles

Explanation:

Step1: Verify volume per marble

Use slope formula: $\text{slope} = \frac{\Delta y}{\Delta x} = \frac{12-10}{1-0} = 2$

Step2: Define volume of marbles

Volume per marble = 2, number of marbles = $x$. So volume of marbles = $2 \cdot x$

Step3: Build volume equation

Substitute into given structure: $v = 10 + 2 \cdot x$

Answer:

The blank for volume per marble is $\boldsymbol{2}$, the "?" for number of marbles is $\boldsymbol{x}$, and the full volume equation is $\boldsymbol{v = 10 + 2x}$