If the surface areas of two spheres are in the ratio of 9: 49, then the ratio of their volumes is:
- A81:311
- B27:143
- C27:343
- D343:81
Solution & Step-by-step Explanation
Let the radii of the two spheres be r
1
and r
2
.
The formula for the surface area of a sphere is 4πr
2
.
Given the ratio of their surface areas:
4πr
2
2
4πr
1
2
=
49
9
(
r
2
r
1
)
2
=
49
9
r
2
r
1
=
49
9
=
7
3
The formula for the volume of a sphere is
3
4
πr
3
.
The ratio of their volumes is:
V
2
V
1
=
3
4
πr
2
3
3
4
πr
1
3
=(
r
2
r
1
)
3
V
2
V
1
=(
7
3
)
3
=
343
27
Therefore, the ratio of their volumes is 27:343.
1
and r
2
.
The formula for the surface area of a sphere is 4πr
2
.
Given the ratio of their surface areas:
4πr
2
2
4πr
1
2
=
49
9
(
r
2
r
1
)
2
=
49
9
r
2
r
1
=
49
9
=
7
3
The formula for the volume of a sphere is
3
4
πr
3
.
The ratio of their volumes is:
V
2
V
1
=
3
4
πr
2
3
3
4
πr
1
3
=(
r
2
r
1
)
3
V
2
V
1
=(
7
3
)
3
=
343
27
Therefore, the ratio of their volumes is 27:343.