If a cone of radius 10.5cm and height 12cm is melted and constructed into a cylinder of the same radius, what will be the height of this cylinder?

When an object is melted and recast into another shape, its total volume remains constant.
Therefore,

Volume of Cylinder=Volume of Cone
The formula for the volume of a cone is:

V
cone
​
=
3
1
​
πr
1
2
​
h
1
​

The formula for the volume of a cylinder is:

V
cylinder
​
=πr
2
2
​
h
2
​

Given:

Radius of cone, r
1
​
=10.5cm

Height of cone, h
1
​
=12cm

Radius of cylinder, r
2
​
=10.5cm (same as cone, so r
1
​
=r
2
​
=r)

Let the height of cylinder be h
2
​
.

Equating the volumes:

πr
2
h
2
​
=
3
1
​
πr
2
h
1
​

Canceling πr
2
from both sides:

h
2
​
=
3
1
​
h
1
​

h
2
​
=
3
1
​
×12=4cm
Thus, the height of the cylinder is 4cm.