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The radius of a solid right circular cylinder is decreased by 20% and its height is increased by 20%. What is the percentage change in its volume?
23.2% decrease
20% decrease
15.4% decrease
No change
23.2% decrease
The volume of a cylinder is proportional to the square of its radius and its height. By applying successive percentage changes, the net effect is a 23.2% decrease in the total volume.
The volume of a cylinder is proportional to the square of its radius and its height. By applying successive percentage changes, the net effect is a 23.2% decrease in the total volume.
Think of stretching a rubber band; if you make it significantly thinner (radius squared impact) while only slightly stretching its length, the total material volume actually shrinks.
Square the small (radius), multiply by the tall (height) to find the change for all.
V=╧Аr2h тАФ Formula for the volume of a right circular cylinder
Net%=(1+100aтАЛ)(1+100bтАЛ)(1+100cтАЛ)тИТ1 тАФ General formula for successive percentage changes
Given volume V=╧Аr2h. If radius r changes by factor (1тИТ0.20)=0.8 and height h changes by factor (1+0.20)=1.2, the new volume VтА▓=╧А(0.8r)2(1.2h)=╧А(0.64r2)(1.2h)=0.768V. The change is 1тИТ0.768=0.232, which corresponds to a 23.2% reduction.
Volume depends quadratically on the radius (r2)
Volume depends linearly on the height (h)
A 20% decrease means multiplying by 0.8
A 20% increase means multiplying by 1.2
Manufacturing engineering for volume estimation
Fluid dynamics in cylindrical pipes
Constant: ╧АтЙИ3.14159
Option B (20% decrease) is a common distractor ignoring the square relationship of the radius.
A is correct тАФ The reduction in radius dominates the volume calculation due to the square factor, leading to a net 23.2% decrease.
Whenever a dimension is squared in a formula, any percentage change applied to it must also be squared in the net effect calculation.