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The ratio of the volumes of two spheres is 27 : 8. Find the ratio of their surface areas.
- a)4 : 9
- b)9 : 4
- c)2 : 3
- d)1 : 3
Correct answer is option 'B'. Can you explain this answer?
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The ratio of the volumes of two spheres is 27 : 8. Find the ratio of t...
Ratio of volumes of spheres of radii r1 and r2 respectively
Similarly,
ratio of surface areas
According to condition
∴ Ratio of surface areas
Similarly,
ratio of surface areas
According to condition
∴ Ratio of surface areas
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The ratio of the volumes of two spheres is 27 : 8. Find the ratio of t...
Understanding Volume and Surface Area of Spheres
The volume (V) of a sphere is given by the formula: V = (4/3)πr³, where r is the radius. The surface area (A) of a sphere is given by: A = 4πr².
Given Ratio of Volumes
The ratio of the volumes of two spheres is 27:8. This can be expressed as:
- V1 : V2 = 27 : 8
Let the radii of the two spheres be r1 and r2. Then, we can set up the equation based on the volume formula:
- (4/3)πr1³ : (4/3)πr2³ = 27 : 8
This simplifies to:
- r1³ : r2³ = 27 : 8
Finding the Ratio of Radii
From the volume ratio, we can find the ratio of the radii by taking the cube root:
- r1/r2 = (27/8)^(1/3) = 3/2
Finding the Ratio of Surface Areas
Now, using the ratio of the radii to find the ratio of surface areas:
- A1 : A2 = 4πr1² : 4πr2²
This simplifies to:
- A1 : A2 = r1² : r2²
Substituting the ratio of the radii:
- (3/2)² : (2/2)² = 9 : 4
Conclusion
Thus, the ratio of the surface areas of the two spheres is:
- 9 : 4
The correct answer is option 'B'.
The volume (V) of a sphere is given by the formula: V = (4/3)πr³, where r is the radius. The surface area (A) of a sphere is given by: A = 4πr².
Given Ratio of Volumes
The ratio of the volumes of two spheres is 27:8. This can be expressed as:
- V1 : V2 = 27 : 8
Let the radii of the two spheres be r1 and r2. Then, we can set up the equation based on the volume formula:
- (4/3)πr1³ : (4/3)πr2³ = 27 : 8
This simplifies to:
- r1³ : r2³ = 27 : 8
Finding the Ratio of Radii
From the volume ratio, we can find the ratio of the radii by taking the cube root:
- r1/r2 = (27/8)^(1/3) = 3/2
Finding the Ratio of Surface Areas
Now, using the ratio of the radii to find the ratio of surface areas:
- A1 : A2 = 4πr1² : 4πr2²
This simplifies to:
- A1 : A2 = r1² : r2²
Substituting the ratio of the radii:
- (3/2)² : (2/2)² = 9 : 4
Conclusion
Thus, the ratio of the surface areas of the two spheres is:
- 9 : 4
The correct answer is option 'B'.
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The ratio of the volumes of two spheres is 27 : 8. Find the ratio of their surface areas.a)4 : 9b)9 : 4c)2 : 3d)1 : 3Correct answer is option 'B'. Can you explain this answer?
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