Study P5 Mathematics Volume of a Liquid - Geniebook

# Volume Of A Liquid

1. Calculate the volume of a liquid in a rectangular container
2. Calculate the volume of liquid needed to fill a container to a certain height

## Volume Recap:

1. $$\text{Volume Of A Cube} = \text{length} \times \text{length} \times \text{length}$$

1. $$\text{Volume Of A Cuboid} = \text{length} \times \text{breadth} \times \text{height}$$

## 1. Calculate the volume of a liquid in a rectangular container

Question 1:

The diagram below shows a rectangular tank filled with some liquid.

Find the volume of the liquid in the tank.

Solution:

Length of cuboid $$= 12 \;cm$$

Breadth of cuboid $$= 5 \;cm$$

Height of liquid in cuboid $$= 4 \;cm$$

Volume of a liquid in cuboid

\begin{align*}​ &= \text{length} \times \text{breadth} \times \text{height} \\ &= 12 \;cm \times 5 \;cm \times 4 \;cm \\ &= 240 \;cm^3 \end{align*}

$$240 \;cm^3$$

Question 2:

In the diagram below, find the volume of the liquid in the rectangular tank.

Solution:

Length of rectangular tank $$= 14 \;m$$

Breadth of rectangular tank $$= 5 \;m$$

Height of liquid in rectangular tank \begin{align*}​\\ &= 10 \;m\, – \,6 \;m \\ &= 4 \;m \end{align*}

Volume of liquid in tank

\begin{align*}​ &= \text{length} \times \text{breadth} \times \text{height} \\ &= 14 \;m \times 5 \;m \times 4 \;m \\ &= 280 \;m^3 \end{align*}

$$280 \;m^3$$

Question 3:

Find the volume of the liquid in the rectangular tank. Express your answer in litres.

Solution:

Length of rectangular tank $$= 20 \;cm$$

Breadth of rectangular tank $$= 8 \;cm$$

Height of liquid in rectangular tank \begin{align*}​ \\ &= 17 \;cm \;–\; 5 \;cm \\ &= 12 \;cm \end{align*}

Volume of liquid in the tank

\begin{align*}​ &= \text{length} \times \text{breadth} \times \text{height} \\ &= 20 \;cm \times 8 \;cm \times 12 \;cm \\ &= 1920 \;cm^3 \\ &= 1.92 \;litres \end{align*}

$$1.92 \;litres$$

Question 4:

Chloe bought some apple juice and poured all of it into $$2$$ rectangular containers. Container $$P$$ measuring $$15 \;cm$$ by $$8 \;cm$$ by $$6 \;cm$$ is \begin{align*}​ \frac {3}{4} \end{align*} filled while Container $$Q$$ measuring $$25 \;cm$$ by $$18 \;cm$$ by $$10 \;cm$$ is \begin{align*}​ \frac {5}{6} \end{align*} filled. What is the volume of apple juice that Chloe bought?

Solution:

Volume of apple juice in Container $$P$$

\begin{align*}​ &= \frac{3}{4}\times \text{length} \times \text{breadth} \times \text{height} \\ &= \frac{3}{4}\times 15 \;cm \times 8 \;cm \times 6 \;cm \\ &= 540 \;cm^3 \\ \end{align*}

Volume of apple juice in Container $$Q$$

\begin{align*}​ &= \frac{5}{6}\times \text{length} \times \text{breadth} \times \text{height} \\ &= \frac{5}{6}\times 25 \;cm \times 18 \;cm \times 10 \;cm \\ &= 3750 \;cm^3 \\ \end{align*}

Total volume of apple juice Chloe bought

$$=$$ Volume of apple juice in Container $$P \;+$$ Volume of apple juice in Container $$Q$$

\begin{align*}​ &= 540 \;cm^3 + 3750 \;cm^3 \\ &= 4290 \;cm^3 \end{align*}

$$4290 \;cm^3$$

Question 5:

A painter mixed some paint to paint a wall. He poured all the paint into two rectangular containers. Container $$X$$ measuring $$55 \;cm$$ by $$48 \;cm$$ by $$30 \;cm$$ was \begin{align*}​ \frac {5}{8} \end{align*} filled while Container $$Y$$ measuring $$47 \;cm$$ by $$39 \;cm$$ by $$30 \;cm$$ was \begin{align*}​ \frac {10}{13} \end{align*} filled. What was the volume of the paint that the painter mixed?

Solution:

Volume of paint in Container $$X$$

\begin{align*}​ &= \frac{5}{8}\times \text{length} \times \text{breadth} \times \text{height} \\ &= \frac{5}{8}\times 55 \;cm \times 48 \;cm \times 30 \;cm \\ &= 49\,500 \;cm^3 \\ \end{align*}

Volume of paint in Container $$Y$$

\begin{align*}​ &= \frac{10}{13}\times \text{length} \times \text{breadth} \times \text{height} \\ &= \frac{10}{13}\times 47 \;cm \times 39 \;cm \times 30 \;cm \\ &= 42\,300 \;cm^3 \\ \end{align*}

Total volume of paint mixed

$$=$$ Volume of paint in Container $$X$$ $$+$$ Volume of paint in Container $$Y$$

\begin{align*}​ &= 49 \,500 \;cm^3 + 42 \,300 cm^3 \\ &= 91 \,800 \;cm^3 \end{align*}

$$91 \,800 \;cm^3$$

## 2. Calculate the volume of liquid needed to fill a container to a certain height

Question 1:

A rectangular tank shown below was half–filled with water. Another $$10.05 \;litres$$ of water was then added to the tank. How much more water was needed to fill the tank to the brim? Give your answer in litres.

Solution:

$$1000 \;cm^3 = 1 \;ℓ$$

Volume of water in the tank

\begin{align*}​ ​&= \frac{1}{2} \times 35 \;cm \times 18 \;cm \times 36 \;cm \\ &= 11 \;340 \;cm^3 \\ &= 11.34 \;ℓ​ \end{align*}

Remaining volume left to fill the tank

\begin{align*}​ ​&​= \frac{1}{2} \times 35 \;cm \times 18 \;cm \times 36 \;cm\\ &= 11 340 \;cm^3\\ &= 11.34 \;ℓ​ \end{align*}

Volume of water added in the tank $$= 10.05 \;ℓ$$

Volume of water needed to fill till the brim

\begin{align*}​ ​&​= 11.34 \;ℓ − 10.05 \;ℓ\\ &= 1.29 \;ℓ \end{align*}

$$1.29 \;ℓ$$

Question 2:

Jenny filled \begin{align*}​ \frac {2}{3} \end{align*} of a square–based container with a cocktail. The guests drank $$2.5 \;litres$$ of cocktail. How much more cocktail must she add in order to fill the container to the brim? Give your answer in litres.

Solution:

$$1000 \;cm^3 = 1 \;ℓ$$

Volume of cocktail

\begin{align*}​ ​&= \frac{1}{2} \times 11 \;cm \times 11 \;cm \times 45 \;cm \\ &= 3630 \;cm^3 \\ &= 3.63 \;ℓ​ \end{align*}

Volume of cocktail drank $$= 2.5 \;ℓ$$

Volume of cocktail left\begin{align*}​\\ &= 3.63 \;ℓ – 2.5 \;ℓ \\ &= 1.13 \;ℓ \end{align*}

Volume of container

\begin{align*}​ &= 11 \;cm \times 11 \;cm \times 45 \;cm \\ &= 5445 \;cm^3 \\ &= 5.445 \;ℓ \end{align*}

Volume of cocktail needed to fill till the brim

\begin{align*}​ &= 5.445 \;ℓ – 1.13 \;ℓ \\ &= 1.13 \;ℓ \end{align*}

$$4.315 \;ℓ$$

Question 3:

Karen filled \begin{align*}​ \frac {1}{3} \end{align*} of rectangular Container $$A$$ measuring $$24 \;cm$$ by $$18 \;cm$$ by $$30 \;cm$$ with orange juice. Karen then added more orange juice such that half of Container $$A$$ was filled with orange juice. How much more orange juice did Karen add to Container $$A$$?

Solution:

Volume of orange juice in the tank

\begin{align*}​ ​&= \frac{1}{3} \times 24 \;cm \times 18 \;cm \times 30 \;cm \\ &= 4320 \;cm^3 \\ \end{align*}

Volume of \begin{align*}​ \frac {1}{2} \end{align*} of Container $$A$$

\begin{align*}​ ​&= \frac{1}{2} \times 24 \;cm \times 18 \;cm \times 30 \;cm \\ &= 6480 \;cm^3 \\ \end{align*}

Volume of orange juice Karen added

\begin{align*}​ &= 6480 \;cm^3 – 4320 \;cm^3 \\ &= 2160 \;cm^3 \end{align*}

$$2160 \;cm^3$$

OR

$$\frac {1}{2}-\frac{1}{3} = \frac{1}{6}$$

Volume of orange juice Karen added

\begin{align*}​ ​&= \frac{1}{6} \times 24 \;cm \times 18 \;cm \times 30 \;cm \\ &= 2160 \;cm^3 \\ \end{align*}

$$2160 \;cm^3$$

## Conclusion

In this article, we have learnt how to calculate the volume of liquids in a container as per the Primary 5 Maths level syllabus.

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