# 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

**Solution:**

A figure is drawn below to visualize the shapes according to the given question.

As it is given that a cylindrical bucket filled with sand is emptied on the ground and a conical heap of sand is formed, the volume of sand in the cylindrical bucket will be the same as the volume of the conical heap of sand.

Therefore, volume of the conical heap of sand = volume of sand in the cylindrical bucket

Let us find the volume of the sand by using formulae;

Volume of the cylinder = πr^{2}h, where r and h are the radius and height of the cylinder respectively

Volume of the cone = 1/3 πr^{2}h,

l = √r^{2} + h^{2}

where r, h, and l are radius, height, and slant height of the cone respectively.

Height of the cylindrical bucket, h₁ = 32 cm

Radius of the cylindrical bucket, r₁ = 18 cm

Height the of conical heap, h = 24 cm

Let the radius of the conical heap be r and slant height be l.

volume the conical heap of sand = volume of sand in the cylindrical bucket

1/3 πr^{2}h = πr₁^{2}h₁

r^{2} = 3r₁^{2}h₁_{ }/ h

r^{2} = [3 × (18 cm)^{2} × 32 cm] /24 cm

r^{2} = (18 cm)^{2} × 4

r = 18 cm × 2 = 36 cm

Slant height, l = √r^{2} + h^{2}

= √(36 cm)^{2} + (24 cm)^{2}

= √1296 cm^{2} + 576 cm^{2}

= √1872 cm^{2}

= 12√13 cm

Thus, the radius and slant height of the conical heap is 36 cm and 12√13 cm respectively.

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 13

**Video Solution:**

## 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

NCERT Solutions for Class 10 Maths Chapter 13 Exercise 13.3 Question 7

**Summary:**

If a cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand and this bucket is emptied on the ground, a conical heap of sand is formed and if the height of the conical heap is 24 cm, the radius and slant height of the heap is 36 cm and 12√13 cm respectively.

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