The height of one cone is 3 times the height of another cone, while its radius is half of the radius of the other cone. If their total volume is 100unit
3
, then the difference in the volumes of the cones is _______ unit
3
.

Let the two cones be Cone 1 and Cone 2.
Let the height of Cone 2 be h and its radius be r.
According to the question, for Cone 1:

Height (h
1
​
)=3h
Radius (r
1
​
)=
2
r
​

The formula for the volume of a cone is V=
3
1
​
πR
2
H.

Volume of Cone 1 (V
1
​
):

V
1
​
=
3
1
​
π(
2
r
​
)
2
(3h)=
3
1
​
π(
4
r
2

​
)(3h)=
4
3
​
(
3
1
​
πr
2
h)
Volume of Cone 2 (V
2
​
):

V
2
​
=
3
1
​
πr
2
h
Let V
2
​
=V. Then V
1
​
=
4
3
​
V.
The total volume is given as 100unit
3
:

V
1
​
+V
2
​
=100
4
3
​
V+V=100
4
7
​
V=100⟹V=
7
400
​

Now, we need to find the difference in their volumes:

Difference=∣V
2
​
−V
1
​
∣=V−
4
3
​
V=
4
1
​
V
Difference=
4
1
​
×
7
400
​
=
7
100
​
≈14.285unit
3
≈14.3unit
3