Volume Of A Liquid
In this article, the objectives are:
- Calculate the volume of a liquid in a rectangular container
- Calculate the volume of liquid needed to fill a container to a certain height
Volume Recap:
- \( \text{Volume Of A Cube} = \text{length} \times \text{length} \times \text{length}\)
- \( \text{Volume Of A Cuboid} = \text{length} \times \text{breadth} \times \text{height}\)
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
\(\begin{align*} &= \text{length} \times \text{breadth} \times \text{height} \\ &= 12 \;cm \times 5 \;cm \times 4 \;cm \\ &= 240 \;cm^3 \end{align*}\)
Answer:
\(240 \;cm^3\)
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
\(\begin{align*} &= \text{length} \times \text{breadth} \times \text{height} \\ &= 14 \;m \times 5 \;m \times 4 \;m \\ &= 280 \;m^3 \end{align*}\)
Answer:
\(280 \;m^3\)
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
\(\begin{align*} &= \text{length} \times \text{breadth} \times \text{height} \\ &= 20 \;cm \times 8 \;cm \times 12 \;cm \\ &= 1920 \;cm^3 \\ &= 1.92 \;litres \end{align*}\)
Answer:
\(1.92 \;litres\)
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 \(Q\)
\(\begin{align*} &= 540 \;cm^3 + 3750 \;cm^3 \\ &= 4290 \;cm^3 \end{align*}\)
Answer:
\(4290 \;cm^3\)
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 \(Y\)
\(\begin{align*} &= 49 \,500 \;cm^3 + 42 \,300 cm^3 \\ &= 91 \,800 \;cm^3 \end{align*}\)
Answer:
\(91 \,800 \;cm^3\)
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
\(\begin{align*} &= 11.34 \;ℓ − 10.05 \;ℓ\\ &= 1.29 \;ℓ \end{align*}\)
Answer:
\(1.29 \;ℓ\)
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
\(\begin{align*} &= 5.445 \;ℓ – 1.13 \;ℓ \\ &= 1.13 \;ℓ \end{align*}\)
Answer:
\(4.315 \;ℓ\)
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
\(\begin{align*} &= 6480 \;cm^3 – 4320 \;cm^3 \\ &= 2160 \;cm^3 \end{align*}\)
Answer:
\(2160 \;cm^3\)
OR
\(\frac {1}{2}-\frac{1}{3} = \frac{1}{6}\)
Volume of orange juice Karen added
\(\begin{align*} &= \frac{1}{6} \times 24 \;cm \times 18 \;cm \times 30 \;cm \\ &= 2160 \;cm^3 \\ \end{align*}\)
Answer:
\(2160 \;cm^3\)
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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