Question

# A vessel is in the form of an inverted cone. Its height is $8cm$ and the radius of its top, which is open, is$5cm.$ It is filled with water up to the brim. When lead shots, each of which is a sphere of radius$0.5cm$ are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

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Solution

## Step1:Given dataHeight of cone $\left(h\right)=8cm$Radius of cone $\left(r\right)=5cm$Radius of sphere $\left(R\right)=0.5cm$Step 2: Find the volume of a cone. The volume of water $=$ Volume of a cone $=\frac{1}{3}\pi {r}^{2}h\phantom{\rule{0ex}{0ex}}=\frac{1}{3}×\pi ×5×5×8\phantom{\rule{0ex}{0ex}}=\frac{200}{3}\pi c{m}^{3}$The volume of water that flows out$=\frac{1}{4}$(total volume of water)$=\frac{1}{4}×\frac{200}{3}\pi =\frac{50}{3}\pi c{m}^{3}$Step 3: Find the volume of the sphere. The volume of each sphere $=\frac{4}{3}\pi {r}^{3}$ $=\frac{4}{3}×\pi ×0.5×0.5×0.5\phantom{\rule{0ex}{0ex}}=\frac{\pi }{6}c{m}^{3}$Step 4: Find the number of lead shots. Let the number of lead shots be $n$Then $n=$$\frac{VolumeofWateroverflown}{VolumeofLeadshot}$ $=\frac{\frac{50}{3}\overline{)\pi }}{\frac{\overline{)\pi }}{6}}\phantom{\rule{0ex}{0ex}}=50×2\phantom{\rule{0ex}{0ex}}=100$$\therefore n=100spheres$Hence, the number of lead shots are $100.$

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