The ratio of the heights of two right circular cones is 25:64 and the ratio of their diameters is 4:5. The ratio of their volumes is
- A1:3
- B1:4
- C5:8
- D1:2
Solution & Step-by-step Explanation
Let the heights of the two cones be h
1
and h
2
, and their radii be r
1
and r
2
.
Given:
Ratio of heights:
h
2
h
1
=
64
25
Ratio of diameters is equal to the ratio of their radii:
r
2
r
1
=
5
4
The volume (V) of a right circular cone is given by:
V=
3
1
πr
2
h
Finding the ratio of their volumes (
V
2
V
1
):
V
2
V
1
=
3
1
πr
2
2
h
2
3
1
πr
1
2
h
1
=(
r
2
r
1
)
2
×(
h
2
h
1
)
Substitute the given ratios into the equation:
V
2
V
1
=(
5
4
)
2
×(
64
25
)
V
2
V
1
=
25
16
×
64
25
V
2
V
1
=
64
16
=
4
1
Hence, the ratio of their volumes is 1:4.
1
and h
2
, and their radii be r
1
and r
2
.
Given:
Ratio of heights:
h
2
h
1
=
64
25
Ratio of diameters is equal to the ratio of their radii:
r
2
r
1
=
5
4
The volume (V) of a right circular cone is given by:
V=
3
1
πr
2
h
Finding the ratio of their volumes (
V
2
V
1
):
V
2
V
1
=
3
1
πr
2
2
h
2
3
1
πr
1
2
h
1
=(
r
2
r
1
)
2
×(
h
2
h
1
)
Substitute the given ratios into the equation:
V
2
V
1
=(
5
4
)
2
×(
64
25
)
V
2
V
1
=
25
16
×
64
25
V
2
V
1
=
64
16
=
4
1
Hence, the ratio of their volumes is 1:4.