Question : Let A and B be two cylinders such that the capacity of A is the same as that of B. The ratio of the diameters of A and B is 1 : 4. What is the ratio of the heights of A and B?
Option 1: 16 : 1
Option 2: 16 : 3
Option 3: 3 : 16
Option 4: 1 : 16
Correct Answer: 16 : 1
Solution :
Capacity of cylinder A = Capacity of cylinder B
The ratio of the diameters of A and B = 1 : 4
⇒ $\frac{r_1}{r_2} = \frac{1}{4}$
⇒ $r_2 = 4r_1$
Let $h_1$ and $h_2$ be the heights of cylinders.
The volume of the cylinder = $\pi× r^2× h$, where $r$ is the radius and $h$ is the height.
So, according to the question,
$\pi× r_1^2 ×h_1 = \pi× r_2^2× h_2$
⇒ $\frac{h_1}{h_2} = \frac{r_2^2}{r_1^2}= \frac{(4r_1)^2}{r_1^2}= \frac{16}{1}$
$\therefore$ Required ratio = 16 : 1
Hence, the correct answer is 16 : 1.
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