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# The length, breadth and height of a solid metallic cuboid are $44cm,21cm$ and $12cm$ respectively. It is melted and a solid cone is made out of it. If the height of the cone is$24cm$, then find the diameter of its base.

Last updated date: 13th Jul 2024
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The problem statement says, the cuboid is melted so the hack here is, volume will not be changed. So, the volume of cuboid will be equal to the volume of solid cone. We know that, Volume of cuboid$= l \times b \times h$where,$l = length,b = breath\;{\text{and }}h = height$. Also, volume of cone$= \dfrac{1}{3}\pi {r^2}h$.
$l \times b \times h = \dfrac{1}{3}\pi {r^2}h \\ \Rightarrow 44 \times 21 \times 12 = \dfrac{1}{3} \times \dfrac{{22}}{7}{r^2} \times 24{\text{ }}[l = 44cm,b = 21cm,{h_{cuboid}} = 12cm,{h_{cone}} = 24cm] \\ \Rightarrow 44 \times 21 = \dfrac{1}{3} \times \dfrac{{22}}{7}{r^2} \times 2 \\ \Rightarrow 21 = \dfrac{1}{3} \times \dfrac{{{r^2}}}{7} \\ \Rightarrow {r^2} = 21 \times 3 \times 7 \\ \Rightarrow {r^2} = 21 \times 21 \\ \Rightarrow r = 21cm \\$
But we need to find diameter. So, we’ll use$d = 2r$. That is$d = 2 \times 21 = 42cm$.
Hence the required diameter of the cone is$42cm$.