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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 conical heap is 24 cm, find the radius and slant height of the heap.

Answer:

By identifying the shapes, we have cone and cylinder. On reshaping from cylindrical to

conical, the volume of sand emptied out remains same.

**Cylinder**

R = 18 cm

H = 32 cm

**Cone (heap)**

r = ?

h = 24 cm

l = ?

∴ Volume of conical heap = Volume of cylinder

\begin{array}{l} \Rightarrow \frac{1}{3} \pi r^{2} h=\pi R^{2} H \\ \Rightarrow \frac{1}{3} r^{2} h=R^{2} H \\ \Rightarrow r^{2}=\frac{3 R^{2} H}{h}=\frac{3 \times 18 \times 18 \times 32}{24} \\ \Rightarrow r^{2}=18 \times 18 \times 2 \times 2 \\ \Rightarrow r=18 \times 2 \mathrm{~cm}=36 \mathrm{~cm} \end{array}

Radius of conical heap is 36 cm.

l^2=r^2+h^2

l^2=36^2+24^2

l^2=1296\ +576

l^2=1872

l=12\sqrt{13}cm

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