Question

Jada uses a graduated cylinder with water in it to measure the volume of some marbles. After dropping in 4 marbles so they are all under water, the water in the cylinder is at a height of 10 milliliters. After dropping in 6 marbles so they are all under water, the water in the cylinder is at a height of 11 milliliters. What will be the height of the water after 13 marbles are dropped

Answer 1

Given that:

- After dropping in 4 marbles, the water in the cylinder is at a height of 10 milliliters.

- After dropping in 6 marbles, the water in the cylinder is at a height of 11 milliliters.

Assuming that the relationship between the height of the water in the cylinder and the number of marble is linear, you can identify these two points on the line:

`(4,10),(6,11)`

You can find the slope of the line using this formula:

`m=(y_2-y_1)/(x_2-x_1)`

Where these two points are on the line:

`(x_1,y_1),(x_2,y_2)`

In this case, you can set up that:

`\begin{gathered} y_2=11 \\ y_1=10 \\ x_2=6 \\ x_1=4 \end{gathered}`

Then, you can determine that:

`m=(11-10)/(6-4)=(1)/(2)`

By definition, the Slope-Intercept Form of the equation of a line is:

`y=mx+b`

Where "m" is the slope of the line and "b" is the y-intercept.

Substitute the slope of the line and the coordinates of one of the points on the line into the equation, and then solve for "b":

`10=((1)/(2))(4)+b`
`10=2+b`
`\begin{gathered} 10-2=b \\ b=8 \end{gathered}`

Now you can write this equation in Slope-Intercept Form, to represent the given situation:

`y=(1)/(2)x+8`

Where "x" is the number of marbles that are dropped in the cylinder and "y" is the height of the water (in milliliters) in the cylinder.

In order to find the height of the water after 13 marbles are dropped, you need to set up that:

`x=13`

Substitute that value into the equation and evaluate:

`y=(1)/(2)(13)+8`
`\begin{gathered} y=(13)/(2)+8 \\ \\ y=14.5 \end{gathered}`

Hence, the answer is:

`14.5\text{ }milliliters`