Question

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 $X$ so that they both have the same volume of water. Is then transferred between $X$ and $Y$ so that they both have the same volume of water. If $h_Y,h_Z$ denote the heights of water at this stage in the buckets $Y,Z,$ respectively, then the ratio $\frac{h_Y}{h_Z}$ equals

• A. $\frac{4}{9}$
• B. $1$
• C. $\frac{9}{4}$
• D. $\frac{81}{40}$

Answer 1

The correct option is D $\frac{81}{40}$


Let initial height of all the buckets $=H$
Volume of bucket $X,Y,Z$ is equal to $\pi H,4\pi H,9\pi H$ respectively.


After first transfer of water between $Z$ and $X$ volume of $Z$ and $X$ is equal it is possible when $\text{volume of}Z=\text{voloume of}X=5\pi H\text{and after first transfer volume of}Y=4\pi H$


After second transfer of water between $X$ and $Y$ volume of $X$ and $Y$ is equal it is possible when $\text{volume of}X=\text{voloume of}Y=4.5\pi H\text{and after second transfer volume of}Z=5\pi H$

$\therefore \text{voloume of}Y=4\pi h_Y=4.5\pi H\frac{h_Y}{H}=\frac{45}{40}$

$\therefore \text{voloume of}Z=9\pi h_Z=5\pi H\frac{h_Z}{H}=\frac{5}{9}$
Hence $\frac{h_Y}{h_Z}=\frac{45\times 9}{40\times 5}=\frac{81}{40}$