Question

A cylindrical bucket, 32 cm high and 18 cm of radius of the base, 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

Height (h1) of cylindrical bucket = 32 cm

Radius (r1) of circular end of bucket = 18 cm

Height (h2) of conical heap = 24 cm

Let the radius of the circular end of conical heap be $r_2$

The volume of sand in the cylindrical bucket will be equal to the volume of sand in the conical heap.

Volume of sand in the cylindrical bucket = Volume of sand in conical heap

$\pi \times r_1^2\times h_1=\frac{1}{3}\pi \times r_2^2\times h_2$

$\pi \times {18}^2\times 32=\frac{1}{3}\pi \times r_2^2\times 24$

${18}^2\times 32=\frac{1}{3}\times r_2^2\times 24$

$r_2^2=\frac{3\times {18}^2\times 32}{24}={18}^2\times 4$

$r_2=18\times 2=36cm$

Slant height = $\sqrt{{36}^2+{24}^2}=\sqrt{{12}^2\times (3^2+2^2)}=12\sqrt{13}cm$

Therefore, the radius and slant height of the conical heap are 36 cm and $12\sqrt{13}$ cm