Find the volume of the smaller cone by halving the volume of the bigger cone in part (b)(i)

$\frac{1}{2} \times 3770 = 1885$ cm³

[1]

Method 1
The ratio of radii (lengths) is 1 : 2 so the ratio of volumes is 1³ : 2³
Split this volume up into the ratio 1 : 8 (by finding $\frac{1}{9}$ of it and $\frac{8}{9}$ of it

$\frac{1}{9} \times 1885 : \frac{8}{9} \times 1885$

The volume of the smaller sphere is $\frac{1}{9} \times 1885$
Form an equation in $r$ by setting the sphere volume formula equal to $\frac{1}{9} \times 1885$

$\frac{4}{3} π r^{3} = \frac{1}{9} \times 1885$

[2]

Get $r^{3}$ on its own (by dividing both sides by $\frac{4}{3} π$

$r^{3} = \frac{\frac{1}{9} \times 1885}{\frac{4}{3} π}$

Find $r$ (by cube-rooting both sides

$r = \sqrt[3]{\frac{\frac{1}{9} \times 1885}{\frac{4}{3} π}}$

3.68 cm [1]

Method 2
The ratio of radii is 1: 2 so write the smaller sphere radius as $r$ and the bigger sphere radius as $2 r$
Form an equation in $r$ (the sum of the volumes of the two spheres equals $1885$ )

$\frac{4}{3} π r^{3} + \frac{4}{3} π {(2 r)}^{3} = 1885$

[1]

Collect like terms, using ${(2 r)}^{3} = 8 r^{3}$

$\frac{4}{3} π r^{3} + \frac{4}{3} π \times 8 r^{3} = 1885 \frac{36}{3} π r^{3} = 1885 12 π r^{3} = 1885$

Find $r$ (by dividing both sides by $12 π$ then cube-rooting)

$r = \sqrt[3]{\frac{1885}{12 π}}$

[1]
3.68 cm [1]

Congruence & Similarity (Cambridge O Level Maths)

Exam Questions