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.

Answer 1

Step1:Given data

Height 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

$$\begin{gathered} =\frac{1}{3}\pi r^{2}h \\ =\frac{1}{3}\times \pi \times 5\times 5\times 8 \\ =\frac{200}{3}\pi c{m}^3 \end{gathered}$$

The volume of water that flows out$=\frac{1}{4}$(total volume of water)

$=\frac{1}{4}\times \frac{200}{3}\pi =\frac{50}{3}\pi c{m}^3$

Step 3: Find the volume of the sphere.

The volume of each sphere $=\frac{4}{3}\pi r^3$

$$\begin{gathered} =\frac{4}{3}\times \pi \times 0.5\times 0.5\times 0.5 \\ =\frac{\pi }{6}c{m}^3 \end{gathered}$$

Step 4: Find the number of lead shots.

Let the number of lead shots be $n$

Then $n=$$\frac{VolumeofWateroverflown}{VolumeofLeadshot}$

$$\begin{gathered} =\frac{{\displaystyle \frac{50}{3}}\cancel{\pi }}{{\displaystyle \frac{\cancel{\pi }}{6}}} \\ =50\times 2 \\ =100 \end{gathered}$$

$\therefore n=100spheres$

Hence, the number of lead shots are $100.$