A cylindrical bucket, 44 cm high and having radius of base 21 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 33 cm, find the radius and the slant height of the heap.

Given.

Height of bucket is 44 cm

Radius of bucket is 21 cm

Height of conical heap is 33 cm

Formula used/Theory.

Volume of cylinder = πr2h

Volume of cone = πr2h

Let the Radius of conical heap be x

Volume of bucket = πr2h

= π × (21 cm)2 × 44 cm

= 19404π cm3

Volume of conical heaps = πr2h

= π × x2 × 33 cm

= 11π × x2 cm

**Note we will not put value of π as it will be divided in next step

As we put bucket of sand on ground it will form a conical heap volume of conical heaps will be equal to volume of bucket

equating both we will get the Radius of conical heap

Volume of bucket = Volume of conical heaps

11π × x2 cm = 19404π cm3

x2 =

x2 = 1764 cm2

x = √ (1764 cm2)

x = 42 cm

Radius of conical heap is 42 cm

In cone;

As the radius , height and slant height makes Right angled triangle where hypotenuse is slant height

Then by Pythagoras theorem

(Slant height)2 = (height)2 + (radius)2

(Slant height)2 = (33 cm)2 + (42 cm)2

(Slant height)2 = 1089 cm2 + 1764 cm2

(Slant height)2 = 2853 cm2

Slant height = √(2853 cm2)

= 53.41 cm

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