Question

A cylindrical bucket, $32$ cm high and with radius of base $18$ 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, then find the radius and slant height of the heap.

Answer 1

Height of cylindrical bucket$\left(h_1\right)=32$ cm

Radius of the base of the bucket $\left(r_1\right)=18$ cm

$\therefore$Volume of the sand in the cylindrical bucket=$\pi r_1^2{h}_1$

Height of conical heap $\left(h_2\right)=24$ cm

let the radius of the conical heap=$r_2$

$\therefore$Volume of the sand in conical heap=$\frac{1}{3}\pi r_2^2{h}_2$

According to the question

The volume of the sand in the cylindrical bucket=Volume of the sand in the conical shape

$\pi r_1^2{h}_1=\frac{1}{3}\pi r_2^2{h}_2$

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

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

$\Rightarrow r_2^2={18}^2\times 4$

$\Rightarrow r_2=18\times 2=36$cm

Slant height of heap=$\sqrt{r_2^2+h_2^2}$

$\Rightarrow \sqrt{{36}^2+{24}^2}$

$\Rightarrow \sqrt{1296+576}$

$\Rightarrow \sqrt{1872}$

$\Rightarrow \sqrt{144\times 13}$

$\Rightarrow 12\sqrt{13}$ cm.