Question

A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied out 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

Height of cylindrical bucket, h = 32 cm
Radius, r = 18 cm
Height of conical heap of sand = 24 cm
Radius, r = ?
Slant height, l = ?
Volume of conical heap of sand = volume of cylinder.
$\frac{1}{3}\pi r^{2}h=\pi r^{2}h$
$\frac{1}{3}\pi \times r^2\times 24=\pi \times (18{)}^2\times 32$
$8r^2=32\times 324$
$r^2=\frac{32\times 324}{8}$
$r^2=1296$
Radius of cone, r = 36 cm
Slant height of cone, l $=\sqrt{h^2+r^2}$
$=\sqrt{(24{)}^2+(36{)}^2}$
$=\sqrt{576+1296}$
$=\sqrt{1872}$
$l=12\sqrt{13}$ cm
$\therefore$ Radius of cone, r = 36 cm
Slant height of cone, h $=12\sqrt{13}$ cm