IB MYP 3 Mathematics 7.5 Capacity, Volume and Measurement Problems Study Notes - New Syllabus

7.5 – Capacity, Volume and Measurement Problems

Volume and Capacity

Volume measures the amount of three-dimensional space occupied by an object. Capacity measures how much a container can hold, usually a liquid.

Volume is normally measured in cubic units, such as \(\text{cm}^3\) and \(\text{m}^3\), while capacity is commonly measured in millilitres and litres.

Connecting Volume and Capacity

There is a direct relationship between cubic centimetres and millilitres:

Another important relationship is:

\(1\text{ m}^3=1000\text{ L}\)

This means that a container with a volume of \(2.5\text{ m}^3\) has a capacity of:

\(2.5\times1000=2500\text{ L}\)

Converting Between Capacity Units

Remember:

\(1\text{ L}=1000\text{ mL}\)

Therefore, to convert:

Example: Convert \(3.75\text{ L}\) to millilitres.

\(3.75\times1000=3750\text{ mL}\)

Therefore, \(3.75\text{ L}=3750\text{ mL}\).

Example: Convert \(4250\text{ mL}\) to litres.

\(4250\div1000=4.25\text{ L}\)

Therefore, \(4250\text{ mL}=4.25\text{ L}\).

Choosing Suitable Units

The unit you choose should match the size of the object being measured.

Finding Capacity from Volume

If the volume of a container is known in \(\text{cm}^3\), it can be converted directly into millilitres.

For example, a container has a volume of \(850\text{ cm}^3\).

\(850\text{ cm}^3=850\text{ mL}\)

In litres:

\(850\div1000=0.85\text{ L}\)

Therefore, the container has a capacity of \(850\text{ mL}\), or \(0.85\text{ L}\).

Finding Volume from Capacity

The process also works in reverse.

If a bottle has a capacity of \(2.4\text{ L}\), then:

\(2.4\times1000=2400\text{ mL}\)

\(2400\text{ mL}=2400\text{ cm}^3\)

Therefore, the container has a volume of \(2400\text{ cm}^3\).

 Measurement Problems with Volume

Many real-life problems require several steps. You may need to:

1. Identify the shape.
2. Find the appropriate volume formula.
3. Calculate the volume.
4. Convert the volume into an appropriate capacity unit.
5. Compare the capacity with the amount available.
6. Interpret the answer in the context of the problem.

Comparing Capacity with an Available Amount

Sometimes a problem gives the capacity of a container and asks whether a certain quantity will fit inside it.

The two quantities must first be expressed in the same units.

For example, suppose a container has capacity \(2.8\text{ L}\), and you have \(2500\text{ mL}\) of liquid.

\(2.8\text{ L}=2800\text{ mL}\)

Since:

\(2500<2800\)

the liquid will fit inside the container.

The unused capacity is:

\(2800-2500=300\text{ mL}\)

Therefore, \(300\text{ mL}\) of space remains.

 Finding an Unknown Dimension from Volume

Sometimes the volume is known, but one of the dimensions is missing. In this situation, use the volume formula and rearrange it.

For a rectangular prism:

\(V=lwh\)

If the height is unknown:

\(h=\frac{V}{lw}\)

For example, a rectangular container has volume \(360\text{ cm}^3\), length \(10\text{ cm}\), and width \(6\text{ cm}\).

\(h=\frac{360}{10\times6}\)

\(h=6\text{ cm}\)

Therefore, the height is \(6\text{ cm}\).

Mixed Units in Measurement Problems

Before calculating volume, all dimensions must be expressed using compatible units.

For example, if a rectangular prism has length \(2\text{ m}\), width \(50\text{ cm}\), and height \(40\text{ cm}\), do not multiply these values immediately.

Convert the dimensions to the same unit first:

\(2\text{ m}=200\text{ cm}\)

Then:

\(V=200\times50\times40\)

\(V=400\,000\text{ cm}^3\)

Since \(1000\text{ cm}^3=1\text{ L}\):

\(400\,000\div1000=400\text{ L}\)

Volume and Capacity Summary