Question

# $504$ cones, each of diameter $3.5\mathrm{cm}$ and height $3\mathrm{cm}$, are melted and recast into a metallic sphere. Find the diameter of the sphere.

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Solution

## Step1 : Find Volume of $504$ conesGiven, Diameter of each cone $=3.5\mathrm{cm}$Radius, $\mathrm{r}=\frac{3.5}{2}$Height of each cone $=3\mathrm{cm}$Thus the volume of each cone, $\mathrm{V}=\frac{1}{3}{\mathrm{\pi r}}^{2}\mathrm{h}$ $=\frac{1}{3}\pi {\left(\frac{3.5}{2}\right)}^{2}×2.3$ $=\frac{1}{3}×\frac{22}{7}×\frac{3.5}{2}×\frac{3.5}{2}×3$ $=\frac{19.25}{2}{\mathrm{cm}}^{3}$Thus, volume of $504$ cones $=504×\frac{19.25}{2}=4851{\mathrm{cm}}^{3}$Step 2: Find the diameter of the sphereAs per the given condition, $504$ cones are melted to form a sphere. Thus, the volume of $504$ cones and sphere made from them will be the same.Hence, Volume of sphere $=4851{\mathrm{cm}}^{3}$ $\frac{4}{3}×\pi ×{\mathrm{r}}^{3}=4851$ ${\mathrm{r}}^{3}=\frac{4851×3}{4×\mathrm{\pi }}$ $\mathrm{r}=10.5\mathrm{cm}$Therefore, the diameter of the sphere is, $\mathrm{d}=2\mathrm{x}10.5=21\mathrm{cm}.$

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