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By what per cent does the curved surface area of a cylinder decrease, if the radius is increased by 30% and the height is decreased by 30\?
0%
30%
60%
9%
9%
Use the formula x+y+100xyтАЛ, where x=+30 and y=тИТ30. The result is 30тИТ30+10030├Ч(тИТ30)тАЛ=тИТ9. The negative sign indicates a 9% decrease.
Radius of the cylinder (r) is increased by 30% and height (h) is decreased by 30%.
Percentage┬аChange=x+y+100x├ЧyтАЛ
Use the formula x+y+100xyтАЛ, where x=+30 and y=тИТ30. The result is 30тИТ30+10030├Ч(тИТ30)тАЛ=тИТ9. The negative sign indicates a 9% decrease.
Students often wrongly assume that increasing one dimension by x% and decreasing another by x% results in a 0% change (no change), ignoring the cross-product term.
Define the formula
The curved surface area (CSA) of a cylinder is given by CSA=2╧Аrh. Since 2╧А is a constant, the CSA is proportional to r├Чh.
CSAтИЭr├Чh
Apply percentage changes
Let initial radius be r and height be h. New radius rтА▓=1.3r and new height hтА▓=0.7h. The new CSA becomes (1.3r)(0.7h)=0.91rh.
0.91├Ч(2╧Аrh)
Calculate the net change
The change factor is 0.91, which represents 91% of the original area. Therefore, the decrease is 100%тИТ91%=9%.
100%тИТ91%=9%┬аdecrease
D is correct because the successive percentage change of +30% and -30% results in a net decrease of 9%.
This concept of 'successive change' is identical to compound interest calculations or successive discount problems, where the order of operations does not change the final product.
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