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If the radius of a sphere is reduced by a factor of 3. Its volume is
- a)reduced by 1/9 of the former volume.
- b)increased by 1/9 of the former volume
- c)reduced by 1/27 of the former volume.
- d)increased by 1/27 of the former volume
Correct answer is option 'C'. Can you explain this answer?
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If the radius of a sphere is reduced by a factor of 3. Its volume isa)...
Reducing the Radius of a Sphere: Explanation of the Solution
Understanding the Problem:
We are given a sphere and are asked to determine how its volume changes when its radius is reduced by a factor of 3.
Formula for the Volume of a Sphere:
The volume of a sphere can be calculated using the formula:
V = (4/3)πr³
where V represents the volume and r represents the radius of the sphere.
Analyzing the Relationship between Radius and Volume:
To understand how the volume of a sphere changes when the radius is altered, we can examine the formula for volume. By isolating the variable r, we can observe the relationship between the two.
V = (4/3)πr³
Dividing both sides by (4/3)π:
V / ((4/3)π) = r³
Taking the cube root of both sides:
(r³)^(1/3) = (V / ((4/3)π))^(1/3)
r = (V / ((4/3)π))^(1/3)
Calculating the New Radius:
If the radius of the sphere is reduced by a factor of 3, the new radius (r') can be calculated as follows:
r' = r / 3 = ((V / ((4/3)π))^(1/3)) / 3
r' = (V / ((4/3)π))^(1/3) / 3
Calculating the New Volume:
To find the new volume (V'), we substitute the new radius (r') into the formula for the volume of a sphere.
V' = (4/3)π(r')³
V' = (4/3)π(((V / ((4/3)π))^(1/3) / 3)³
V' = (4/3)π((V / ((4/3)π))^(1/3))³ / 3³
V' = (4/3)π(V / ((4/3)π)) / 27
V' = V / 27
Conclusion:
Therefore, when the radius of a sphere is reduced by a factor of 3, its volume is reduced by 1/27 of the former volume. Hence, the correct answer is option 'C'.
Understanding the Problem:
We are given a sphere and are asked to determine how its volume changes when its radius is reduced by a factor of 3.
Formula for the Volume of a Sphere:
The volume of a sphere can be calculated using the formula:
V = (4/3)πr³
where V represents the volume and r represents the radius of the sphere.
Analyzing the Relationship between Radius and Volume:
To understand how the volume of a sphere changes when the radius is altered, we can examine the formula for volume. By isolating the variable r, we can observe the relationship between the two.
V = (4/3)πr³
Dividing both sides by (4/3)π:
V / ((4/3)π) = r³
Taking the cube root of both sides:
(r³)^(1/3) = (V / ((4/3)π))^(1/3)
r = (V / ((4/3)π))^(1/3)
Calculating the New Radius:
If the radius of the sphere is reduced by a factor of 3, the new radius (r') can be calculated as follows:
r' = r / 3 = ((V / ((4/3)π))^(1/3)) / 3
r' = (V / ((4/3)π))^(1/3) / 3
Calculating the New Volume:
To find the new volume (V'), we substitute the new radius (r') into the formula for the volume of a sphere.
V' = (4/3)π(r')³
V' = (4/3)π(((V / ((4/3)π))^(1/3) / 3)³
V' = (4/3)π((V / ((4/3)π))^(1/3))³ / 3³
V' = (4/3)π(V / ((4/3)π)) / 27
V' = V / 27
Conclusion:
Therefore, when the radius of a sphere is reduced by a factor of 3, its volume is reduced by 1/27 of the former volume. Hence, the correct answer is option 'C'.
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Community Answer
If the radius of a sphere is reduced by a factor of 3. Its volume isa)...
Reduced by 1/27 of the former volume, Option C.
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If the radius of a sphere is reduced by a factor of 3. Its volume isa)reduced by 1/9 of the former volume.b)increased by 1/9 of the former volumec)reduced by 1/27 of the former volume.d)increased by 1/27 of the former volumeCorrect answer is option 'C'. Can you explain this answer?
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