Question

Two cylinders have equal volume. Their heights are in the ratio $1:2$. Find the ratio of their radii.

Answer 1

Hint: Given that there are two cylinders such that their volumes are same but heights are different. We have to find the ratio of their radii. We will use the formula of volume of the cylinder to solve this problem. We will equate the volumes but will consider the dimension to be different.

Complete step by step answer:
Given that there are two cylinders such that their volumes. Let $R$ and $r$ be the radii of the two cylinders. And $H$ and $h$ be the heights of them.As we know that the volume of a cylinder is given by \[\pi {r^2}h\] we can equate their volumes as,
\[\pi {R^2}H = \pi {r^2}h\]

Cancelling pi from both sides and taking the ratio as,
\[\dfrac{H}{h} = \dfrac{{{r^2}}}{{{R^2}}}\]
But their heights are in the ratio 1:2
\[\dfrac{1}{2} = \dfrac{{{r^2}}}{{{R^2}}}\]
Taking roots on both sides we get,
\[\dfrac{r}{R} = \dfrac{1}{{\sqrt 2 }}\]
$\therefore \dfrac{R}{r} = \dfrac{{\sqrt 2 }}{1}$

Therefore, the ratio of radii of both the cylinders will be \[\sqrt 2 :1\].

Note: Though the volume is the same the dimensions are different. Note that if height is more then definitely radius will be smaller and vice versa because the volume should be the same. So the cylinder with larger height has smaller radius and one with smaller height has larger radius as compared. Do observe the dimensional ratio carefully.