 # NCERT Exemplar Solutions Class 10 Mathematics Solutions for Surface Areas and Volumes - Exercise 12.4 in Chapter 12 - Surface Areas and Volumes

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.

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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