Using Equalities

We can express volume in different ways. For example if you have a can of soda, you could use the formula for the volume of a cylinder and measure the diameter and height of the can. You could also directly read the volume from the label of the can.

The estimated formula and the stated volume should be very close to equal.

Now, imagine you pour that soda into another container. The shape it makes in that container will be different than the can. Most of the time, you won’t know one of the dimensions, but you can still write a formula for the volume.

You can set the known volume to this imaginary volume and then use algebra to find the unknown dimension. Often, this dimension is the depth of the liquid.

Relationship between Depth and volume

On your graph:

V = slope \cdot h

From theory we know that

V = l \cdot w \cdot h A = l \cdot w

Plugging in our formulas above

V = A \cdot h

Other Examples

  • The total area of compost to be delivered equals the volume of one truck times the number of trucks.
  • The formula estimate of the water in a kiddie pool is equal to the gallons in the pool.

Finding the depth of water

Suppose we have a known volume of water that we want to pour in a container. We want to know how deep the water will be in that container. Imagine a container with a base of 15 cm by 13 cm.

We take an algebra approach

  • find an equivalency
    • volume in beaker
    • volume of our model of the water poured into the box
  • express each side differently
    • one side: 700 ml water in beaker
    • other side: an imaginary box of 15 cm x 13 cm x h
  • use algebra rules to create insight

\textrm{imaginary volume in box} = \textrm{volume measured in beaker} l \cdot w \cdot h = V_{beaker}

We can directly measure l, h, and V_{beaker} and plug them in and solve for h. As we do this, we need to certain of our unit conversions. For example, 1 milliliter is equal to one cubic centimeter.

15 cm \cdot 13 cm \cdot h = 700 ml 15 cm \cdot 13 cm \cdot h = 700 cm^3 13 cm \cdot h = \frac{700 cm^3}{15 cm} h = \frac{700 cm^3}{15 cm \cdot 13 cm} = 3.6 cm