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.

First, find the slope of the line.

x

y

0

1

2

3

4

5

0

2

4

6

8

10

12

14

16

Number of marbles, x

Volume in mL, v

slope : 

4

1234567890.−

Enter

Answer 1

To find the equation that represents the relationship between the number of marbles (x) and the volume in a cylinder (v), we first need to determine the slope of the line from the data points provided in the graph.

From the data:
- At \( x = 0, v = 0 \)
- At \( x = 1, v = 2 \)
- At \( x = 2, v = 4 \)
- At \( x = 3, v = 6 \)
- At \( x = 4, v = 8 \)
- At \( x = 5, v = 10 \)

It looks like the relationship is linear. We can calculate the slope (m) using two points, for example, from the points \( (1, 2) \) and \( (0, 0) \).

The formula for slope \( m \) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points \( (0, 0) \) and \( (1, 2) \):

\[
m = \frac{2 - 0}{1 - 0} = \frac{2}{1} = 2
\]

To confirm the consistency, we can also use other points:
Using points \( (2, 4) \) and \( (1, 2) \):

\[
m = \frac{4 - 2}{2 - 1} = \frac{2}{1} = 2
\]

This confirms that the slope of the line is indeed \( 2 \).

Now, we can use the slope-intercept form of a line, which is:

\[
v = mx + b
\]

Where:
- \( m \) is the slope
- \( b \) is the y-intercept

Since the y-intercept (v when x=0) is \( 0 \):

\[
b = 0
\]

Thus, the equation representing the relationship will be:

\[
v = 2x
\]

This equation describes the volume in mL as a function of the number of marbles.