Question

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

Answer 1

Height (h) of conical vessel = 8 cm
Radius (r₁) of conical vessel = 5 cm
Radius (r₂) of lead shots = 0.5 cm

Let n number of lead shots were dropped in the vessel.

Volume of water spilled = Volume of dropped lead shots 
Total volume of water overflown= \((\frac{1}{4})×(\frac{200}{3}) \pi =(\frac{50}{3})\pi\)

The volume of lead shot
\(= (\frac{4}{3})\pi r^3\)
\(= (\frac{1}{6}) \pi\)

\(\text{The number of lead shots}=\frac{\text{Total volume of water overflown}}{\text{Volume of lead shot}}\)

\(=\frac{ (\frac{50}{3})\pi }{(\frac{1}{6})\pi}\)

\(= (\frac{50}{3})\times6\)
\(=100\)

Answer 2

Step 1: Volume of the Cone
\(V = \frac{1}{3} \pi r^2 h\)
Given: r = 5cm and h = 8cm
Substituting the values:
\(V = \frac{1}{3} \pi (5)^2 (8)\)

\(V = \frac{1}{3} \pi (25) (8)\)

\(V = \frac{1}{3} \pi (200)\)

\(V = \frac{200}{3} \pi \, \text{cm}^3\)

Step 2: Volume of Water that Flows Out
One-fourth of the water flows out, so the volume of water that flows out is:
\(V_{\text{out}} = \frac{1}{4} \times \frac{200}{3} \pi\)

\(V_{\text{out}} = \frac{200 \pi}{12}\)

\(V_{\text{out}} = \frac{50 \pi}{3} \, \text{cm}^3\)

Step 3: Volume of One Lead Shot
The volume V of a sphere is given by the formula: \(V = \frac{4}{3} \pi r^3\)
Given the radius of each lead shot is 0.5cm:
\(V_{\text{shot}} = \frac{4}{3} \pi (0.5)^3\)

\(V_{\text{shot}} = \frac{4}{3} \pi (0.125)\)

\(V_{\text{shot}} = \frac{4}{24} \pi\)

\(V_{\text{shot}} = \frac{1}{6} \pi \, \text{cm}^3\)

Step 4: Number of Lead Shots
To find the number of lead shots n, we divide the total volume of water that flows out by the volume of one lead shot:
\(n = \frac{V_{\text{out}}}{V_{\text{shot}}}\)

\(n = \frac{\frac{50 \pi}{3}}{\frac{1}{6} \pi}\)

Simplify the expression:
\(n = \frac{50 \pi}{3} \times \frac{6}{\pi}\)

\(n = \frac{50 \times 6}{3}\)

\(n = \frac{300}{3}\)
\(n = 100\)

So, the correct answer is 100.