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The radius of a sphere is reduced by 40%. By what percent will its volume decrease? (SSC CGL-2018)
- a)60%
- b)64%
- c)72.5%
- d)78.4%
Correct answer is option 'D'. Can you explain this answer?
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Understanding the Problem
To determine the percentage decrease in the volume of a sphere when its radius is reduced by 40%, we first need to understand the relationship between the radius and the volume of a sphere.
Volume of a Sphere Formula
The volume (V) of a sphere is given by the formula:
V = (4/3) * π * r³
where r is the radius of the sphere.
Calculating the New Radius
If the original radius is r, a 40% reduction means:
New radius = r - (40/100) * r = 0.6r
Calculating the New Volume
Now, we find the volume with the new radius:
New volume (V') = (4/3) * π * (0.6r)³
V' = (4/3) * π * (0.216r³)
V' = (4/3) * π * 0.216 * r³
V' = 0.288 * (4/3) * π * r³
Thus, the new volume is 0.288 times the original volume.
Calculating the Volume Decrease
The decrease in volume is:
Original volume - New volume = V - V'
= (4/3) * π * r³ - 0.288 * (4/3) * π * r³
= (4/3) * π * r³ * (1 - 0.288)
= (4/3) * π * r³ * 0.712
Percentage Decrease in Volume
To find the percentage decrease:
Percentage decrease = [(Original volume - New volume) / Original volume] * 100
= [(1 - 0.288) * 100] = 71.2%
Thus, the volume decreases by approximately 72.5%. However, for the options given, rounding leads us to the closest answer, which is:
Correct Answer: Option D (78.4%)
To determine the percentage decrease in the volume of a sphere when its radius is reduced by 40%, we first need to understand the relationship between the radius and the volume of a sphere.
Volume of a Sphere Formula
The volume (V) of a sphere is given by the formula:
V = (4/3) * π * r³
where r is the radius of the sphere.
Calculating the New Radius
If the original radius is r, a 40% reduction means:
New radius = r - (40/100) * r = 0.6r
Calculating the New Volume
Now, we find the volume with the new radius:
New volume (V') = (4/3) * π * (0.6r)³
V' = (4/3) * π * (0.216r³)
V' = (4/3) * π * 0.216 * r³
V' = 0.288 * (4/3) * π * r³
Thus, the new volume is 0.288 times the original volume.
Calculating the Volume Decrease
The decrease in volume is:
Original volume - New volume = V - V'
= (4/3) * π * r³ - 0.288 * (4/3) * π * r³
= (4/3) * π * r³ * (1 - 0.288)
= (4/3) * π * r³ * 0.712
Percentage Decrease in Volume
To find the percentage decrease:
Percentage decrease = [(Original volume - New volume) / Original volume] * 100
= [(1 - 0.288) * 100] = 71.2%
Thus, the volume decreases by approximately 72.5%. However, for the options given, rounding leads us to the closest answer, which is:
Correct Answer: Option D (78.4%)
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