Question : The radius of a cylinder is 10 cm and the height is 4 cm. The number of centimeters that may be added either to the radius or to the height to get the same increase in the volume of the cylinder is:
Option 1: 5 cm
Option 2: 4 cm
Option 3: 25 cm
Option 4: 16 cm
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Correct Answer: 5 cm
Solution :
The radius of a cylinder = 10 cm
Height = 4 cm
Let the radius be increased by $x$ cm.
⇒ Volume of cylinder = $\pi(10+x)^{2}×4$----------(i)
Let height be increased by $x$ cm.
⇒ Volume of cylinder = $\pi×10^{2}×(4+x)$-------(ii)
From equations (i) and (ii),
$\pi(10+x)^{2}×4$ = $\pi×10^{2}×(4+x)$
⇒ $(10+x)^{2}×4$ = $10^{2}×(4+x)$
⇒ $(100+x^{2}+20x)×4$ = $100×(4+x)$
⇒ $4x^{2}+80x$ = $100x$
⇒ $4x^{2}$ = $20x$
⇒ $x = 5$
Hence, the correct answer is 5 cm.
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