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If the radius of sphere is decreased by 10 percent, then the volume of sphere will be decreased by what percentage?
25.6%
32.5%
30.6%
27.1%
27.1%
Use the successsive percentage change formula for volume (3 dimensions): x+x+x+100x2+x2+x2тАЛ+10000x3тАЛ, where x=тИТ10. Or simply: (0.9)3=0.729, which is a 1тИТ0.729=0.271 or 27.1% decrease.
Radius of a sphere is decreased by 10%.
V=34тАЛ╧Аr3,Percentage┬аchange=(1тИТ(1тИТ100xтАЛ)3)├Ч100
Use the successsive percentage change formula for volume (3 dimensions): x+x+x+100x2+x2+x2тАЛ+10000x3тАЛ, where x=тИТ10. Or simply: (0.9)3=0.729, which is a 1тИТ0.729=0.271 or 27.1% decrease.
Many students mistake the volume decrease for 30% (applying 10%├Ч3) or miscalculate (0.9)3 as 0.81 (which is 0.92).
Define Initial and Final Radius
Let the initial radius be r. If the radius is decreased by 10%, the new radius rтА▓ is rтИТ0.10r=0.9r.
rтА▓=0.9r
Calculate Volume Change Ratio
The volume of a sphere is V=34тАЛ╧Аr3. The ratio of the new volume VтА▓ to the original volume V is given by the cube of the radius ratio: VтА▓/V=(rтА▓/r)3=(0.9)3.
(0.9)3=0.729
Calculate Percentage Decrease
The percentage decrease is (1тИТ0.729)├Ч100%=0.271├Ч100%=27.1%.
27.1%
D is correct because the volume of a sphere is proportional to the cube of its radius, and (0.9)3=0.729, representing a decrease of 1тИТ0.729=0.271 or 27.1%.
This concept of cubing the change factor is identical to calculating compound interest over 3 periods or calculating volume changes for cubes where side length changes.
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