Given are three cylindrical buckets $X, Y, Z$ whose circular bases are of radii $1,2,3$ units, respectively, Initially water is filled in these buckets upto the same height. Some water is then transferred from $Z$ to $Z$ so that they both have the same volume of water. Some water is then transferred between $X$ and $Y$ so that they both have the same volume of water. If $\mathrm{h}_{\mathrm{y}}, \mathrm{h}_{\mathrm{z}}$ denote the heights of water at this stage in the buckets $\mathrm{Y}, \mathrm{Z}$, respectively, then the ratio $\frac{h_{y}}{h_{z}}$ equals.

• A

  $\frac{4}{9}$
• B

  1
• C

  $\frac{9}{4}$
• Correct answer
  D

  $\frac{81}{40}$

Solution

$\begin{array}{lllll}\mathrm{V}= & \pi \mathrm{h} & \pi 4 \mathrm{~h} & \pi 9 \mathrm{~h} & \ldots \ldots \text { stage (1) } \\ & 5 \pi \mathrm{h} & 4 \pi \mathrm{h} & 5 \pi \mathrm{h} & \ldots \ldots \text { stage (2) } \\ & 4.5 \pi \mathrm{h} & 4.5 \pi \mathrm{h} & 5 \pi \mathrm{h} & \ldots \ldots \text { stage (3) } \\ 4 \pi \cdot \mathrm{h}_{\mathrm{y}} & =\frac{9 \pi \mathrm{h}}{2} & \pi 9 \mathrm{~h}_{\mathrm{z}}=5 \pi \mathrm{h} & \end{array}$ $\mathrm{h}_{\mathrm{y}}=\frac{9 \mathrm{~h}}{8} \quad \mathrm{~h}_{\mathrm{z}}=\frac{5}{9} \mathrm{~h}$ $\frac{\mathrm{h}_{\mathrm{y}}}{\mathrm{h}_{\mathrm{z}}}=\frac{9 \mathrm{~h}}{8} / \frac{5 \mathrm{~h}}{9}$ $=\frac{81}{40}$

Hence, the correct answer is option: 'D'.