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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, find the radius and slant height of the heap.

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Radius =27 cm; Slant height=43cm

Radius =27 cm; Slant height=43.26cm

Radius =36 cm; Slant height= 43.16 cm

Radius =36 cm; Slant height= 43.26 cm

answer is D.

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Detailed Solution

It is given height of a cylindrical bucket = 32 cm and radius of the base = 18 cm.
Volume of cylinder =

π r^{2} h

Volume of cone =

\frac{1}{3} π r^{2} h

Where, r = radius and h = height
Since the volume of sand remains constant, we can equate the two volumes

\Rightarrow Volume o f cylinder = volume of cone

\Rightarrow π {\times 18}^{2} \times 32 = \frac{1}{3} π \times r^{2} \times 24

\Rightarrow r^{2} = \frac{18^{2} \times 32}{8}

\Rightarrow r^{2} = 18^{2} \times 2^{2}

\Rightarrow r = 18 \times 2 = 36 cm

Slant height of a cone = l

l^{2} = r^{2} + h^{2}

\Rightarrow l^{2} = 36^{2} + 24^{2}

\Rightarrow l^{2} = 1296 + 576

\Rightarrow l^{2} = 1872

\Rightarrow l = 43.26 cm

Hence option (4) is correct.

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