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The ratio between the volume of two sphere is 8 : 27. What is the ratio between their surface areas?
- a)4 : 9
- b)4 : 5
- c)5 : 6
- d)4 : 7
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
The ratio between the volume of two sphere is 8 : 27. What is the rati...
Let radius o f two spheres be r1, and r2 re spectively .
Given, ratio of volumes = 8 : 27
⇒ 
⇒ r1 : r2 = 2 : 3
Ratio between surface areas = 
= 4/9
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The ratio between the volume of two sphere is 8 : 27. What is the rati...
Given:
The ratio between the volume of two spheres is 8: 27.
To find:
The ratio between their surface areas.
Solution:
Let's consider two spheres with volumes V1 and V2, and surface areas S1 and S2 respectively.
Step 1:
We know that the volume of a sphere is given by the formula:
V = (4/3)πr³, where r is the radius of the sphere.
So, the given ratio between the volumes can be written as:
V1/V2 = 8/27
Step 2:
Let's assume the radii of the two spheres as r1 and r2 respectively.
Therefore, we can write the volumes as:
V1 = (4/3)πr1³
V2 = (4/3)πr2³
Step 3:
Now, let's simplify the given ratio using the expressions for volumes:
(4/3)πr1³ / (4/3)πr2³ = 8/27
The π and (4/3) terms cancel out on both sides, leaving us with:
(r1/r2)³ = 8/27
Step 4:
Taking the cube root of both sides, we get:
r1/r2 = 2/3
Step 5:
We know that the surface area of a sphere is given by the formula:
S = 4πr²
Substituting the radii r1 and r2, we can write the surface areas as:
S1 = 4πr1²
S2 = 4πr2²
Step 6:
Now, let's find the ratio between the surface areas:
S1/S2 = (4πr1²) / (4πr2²)
S1/S2 = (r1²/r2²)
Step 7:
Using the ratio r1/r2 = 2/3 from Step 4, we can substitute it into the above expression:
S1/S2 = (2/3)²
S1/S2 = 4/9
Conclusion:
Therefore, the ratio between the surface areas of the two spheres is 4:9.
Hence, the correct answer is option A) 4:9.
The ratio between the volume of two spheres is 8: 27.
To find:
The ratio between their surface areas.
Solution:
Let's consider two spheres with volumes V1 and V2, and surface areas S1 and S2 respectively.
Step 1:
We know that the volume of a sphere is given by the formula:
V = (4/3)πr³, where r is the radius of the sphere.
So, the given ratio between the volumes can be written as:
V1/V2 = 8/27
Step 2:
Let's assume the radii of the two spheres as r1 and r2 respectively.
Therefore, we can write the volumes as:
V1 = (4/3)πr1³
V2 = (4/3)πr2³
Step 3:
Now, let's simplify the given ratio using the expressions for volumes:
(4/3)πr1³ / (4/3)πr2³ = 8/27
The π and (4/3) terms cancel out on both sides, leaving us with:
(r1/r2)³ = 8/27
Step 4:
Taking the cube root of both sides, we get:
r1/r2 = 2/3
Step 5:
We know that the surface area of a sphere is given by the formula:
S = 4πr²
Substituting the radii r1 and r2, we can write the surface areas as:
S1 = 4πr1²
S2 = 4πr2²
Step 6:
Now, let's find the ratio between the surface areas:
S1/S2 = (4πr1²) / (4πr2²)
S1/S2 = (r1²/r2²)
Step 7:
Using the ratio r1/r2 = 2/3 from Step 4, we can substitute it into the above expression:
S1/S2 = (2/3)²
S1/S2 = 4/9
Conclusion:
Therefore, the ratio between the surface areas of the two spheres is 4:9.
Hence, the correct answer is option A) 4:9.
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