Problem Solving with Volumes (Edexcel IGCSE Maths)

Spec point — 4.11C

Problem-solving with volumes

What is problem-solving?

• Problem solving, usually has two key features:

  • A question is given as a real-life scenario

    • eg. The volume of water in a swimming pool...

  • There is usually more than one topic of maths you will need in order to answer the question

    • eg. Volume and money

What are common problems that involve volume?

• Volume is a commonly used topic of 'real-world' maths

  • For example, a carton of juice in the shape of a cuboid, a cylindrical tin and a triangular prism chocolate box all involve volume

• Typically, the 'real-world' scenarios also have a cost

  • A lot of volume problems also involve calculations with money

How do I solve problems involving volume?

• Often the 3D object in a question will not be a standard cuboid, cone, sphere, etc.

  • It will likely either be:

    • A prism (3D shape with the same cross-section running through it)
    • A portion or fraction of a standard shape (a hemisphere for example)
    • A compound object (an object made up of two or more standard 3D objects)

• If the object is a prism, recall that the volume of a prism is the cross-sectional area × its length

  • The cross-sectional area may be a compound 2D shape

    • For example, an L-shape, or a combination of a rectangle and a triangle

• If the object is a fraction of a standard shape, consider the "full" version of the object and find the appropriate fraction of it

  • A hemisphere is half a sphere

  • A frustum is a truncated(chopped-off) cone

    • The volume of a frustum will be the volume of the smaller cone subtracted from the volume of the larger cone

• If the object is a compound object, find the volumes of the individual standard 3D objects and add them together

• Problem solving questions could appear on either a non-calculator paper or a calculator paper

Exam Hint

• Before you start calculating, make a quick note of your plan to tackle the question

  • For example, "Find the area of the triangle and the rectangle, add together, multiply by the length"

Worked Example The diagram shows a prism.

Work out the volume of the prism.

Answer:

The volume is the area of the cross section × length (10 cm) Find the area by splitting into a 7 × 4 and a (9 - 4) × 2 rectangle (or a 9 × 2 and a (7 - 2) × 4 rectangle)

• 7 × 4 + (9 - 4) × 2 = 38 cm²*

Find the volume (by multiplying 38 by 10)

38 × 10

Final answer: 380 cm³

Worked Example The diagram shows a truncated cone (a frustum). Using the given dimensions, find the volume of the frustum to 3 significant figures.

Answer:

To find the volume of the frustum, find the volume of the larger cone (30 cm tall, with a radius of 20 cm), and subtract the volume of the smaller cone (15 cm tall, with a radius of 10 cm)

Formula for the volume of a cone: $\frac{1}{3}πr^{2}h$

Calculate the volume of the larger cone

$V_{L}=\frac{1}{3}\timesπ\times20^{2}\times30=4000π=12566.37061...\mathrm{cm}^{3}$

Calculate the volume of the smaller cone

$V_{S}=\frac{1}{3}\timesπ\times10^{2}\times15=500π=1570.796327...\mathrm{cm}^{3}$

Find the difference

$V_{L}-V_{S}=4000π-500π=3500π=10995.57429...\mathrm{cm}^{3}$

Round to 3 significant figures

Final answer: 11 000 cm³