Question : If the radius of a cylinder is decreased by 12%, then by how much percentage must its height be increased so that the volume of the cylinder remains the same?
Option 1: 29.13%
Option 2: 21.78%
Option 3: 42.56%
Option 4: 34.27%
Correct Answer: 29.13%
Solution :
Let's denote the original radius of the cylinder as $r$ and the original height as $ℎ$.
Now, if the radius is decreased by 12%,
⇒ $r$′ = $r$ −0.12$r$ = 0.88$r$
⇒ $V'$ = $π(0.88r)^2h'$
Since the volume remains the same
⇒ $πr^{2}h$ = $π(0.88r)^2h'$
⇒ $h'$ = $\frac{r^{2}h}{(0.88r)^2}$
⇒ $h'$ = $\frac{r^{2}h}{(0.7744)r^2}$
⇒ $h'$ = $\frac{100h}{0.7744}$ – $h$
⇒ $Δℎ$ = $h'−h$
⇒ $Δℎ$ = $\frac{100h}{0.7744}$ – $h$
⇒ Percentage change in height = $\frac{Δh}{h}×100$
= $(\frac{100}{77.44} – 1)×100$
= 29.13%
Hence, the correct answer is 29.13%.
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