Question

Question 13
A cylindrical bucket of height 32 cm and base radius 18 cm is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap.

Answer 1

Given, radius of the base of the bucket = 18 cm.
Height of the bucket = 32 cm

So , volume of the sand in cylindrical bucket
$=\pi r^{2}h=\pi (18{)}^2\times 32=10368\pi$

Also, given height of the conical heap (h) = 24 cm
Let radius of heap be r cm.

Then, volume of the sand in the heap
$=\frac{1}{3}\pi r^{2}h$
$=\frac{1}{3}\pi r^2\times 24=8\pi r^2$

According to the question.
Volume of the sand in cylindrical bucket = Volume of the sand in conical heap

$\Rightarrow 10368\pi =8\pi r^2\Rightarrow 10368=8r^2\Rightarrow r^2=\frac{10368}{8}=1296\Rightarrow r=36cm$

Again, let the slant height of the conical heap = $l$
Now,
$l^2=h^2+r^2=(24{)}^2+(36{)}^2=576+1296=1872$
$\Rightarrow l=43.267cm$

$\therefore$ Hence, radius of conical heap of sand = 36 cm
and slant height of conical heap = 43.267 cm