If the surface areas of two spheres are in the ratio of 4:25, then find the ratio of their volumes.
- A5:2
- B125:8
- C8:125
- D4:25
Solution & Step-by-step Explanation
Let the radii of the two spheres be R
1
and R
2
.
The surface area of a sphere is given by 4πR
2
.
The ratio of their surface areas is:
4πR
2
2
4πR
1
2
=
25
4
(
R
2
R
1
)
2
=
25
4
⟹
R
2
R
1
=
25
4
=
5
2
The volume of a sphere is given by
3
4
πR
3
.
The ratio of their volumes is:
3
4
πR
2
3
3
4
πR
1
3
=(
R
2
R
1
)
3
Substitute the value of
R
2
R
1
:
Ratio of volumes=(
5
2
)
3
=
125
8
1
and R
2
.
The surface area of a sphere is given by 4πR
2
.
The ratio of their surface areas is:
4πR
2
2
4πR
1
2
=
25
4
(
R
2
R
1
)
2
=
25
4
⟹
R
2
R
1
=
25
4
=
5
2
The volume of a sphere is given by
3
4
πR
3
.
The ratio of their volumes is:
3
4
πR
2
3
3
4
πR
1
3
=(
R
2
R
1
)
3
Substitute the value of
R
2
R
1
:
Ratio of volumes=(
5
2
)
3
=
125
8