# 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.

As it is given that a cylindrical bucket filled with sand, emptied on the ground and a conical heap of sand is formed. Then 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 cone respectively.

Height of the cylindrical bucket, h_{1} = 32 cm

Radius of the cylindrical bucket, r_{1} = 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 = πr_{1}^{2}h_{1}

r^{2} = 3πr_{1}^{2}h_{1 }/ 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 are 36 cm and 12√13 cm respectively.

**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 :

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

The radius and slant height of the conical heap formed by emptying a cylindrical bucket of sand with a height of 32 cm and a base radius of 18 cm are 36 cm and 12√13 cm respectively