Question

A girl fills a cylindrical bucket 32 cm in height and 18 cm in radius with sand. She empties the bucket on the ground and makes a conical heap of the sand. If the height of the conical heap is 24 cm, find :
(i) its radius,
(ii) its slant height. [3 MARKS]

Answer 1

Concept: 1 Mark
Application: 2 Marks

Height of cylindrical bucket, H = 32 cm.

Radius of cylindrical bucket, R = 18 cm.

Volume of sand
$=\pi R^{2}H=(\frac{22}{7}\times 18\times 18\times 32)c{m}^3$ .

(i) Height of conical heap, h = 24 cm

Let the radius of the conical heap be r cm.

Then, volume of conical heap
$=\frac{1}{3}\pi r^{2}h=(\frac{1}{3}\times \frac{22}{7}\times r^2\times 24)c{m}^3$

Now, Volume of conical heap = volume of sand

$\Rightarrow (\frac{1}{3}\times \frac{22}{7}\times r^2\times 24)=(\frac{22}{7}\times 18\times 18\times 32)$

$\Rightarrow r^2=\left(\frac{18\times 18\times 32\times 3}{24}\right)=(18\times 18\times 4)$

$\Rightarrow r=\sqrt{18\times 18\times 4}=(18\times 2)cm=36cm$

$\therefore$ Radius of the heap = 36 cm


(ii) Slant height,
$l=\sqrt{h^2+r^2}=\sqrt{(24{)}^2+(36{)}^2}$

$=\sqrt{1872}$

$=12\sqrt{13}cm$