Question

Assume that a drop of water is spherical and its diameter is one tenth of a cm. A conical glass has equal height to its diameter of rim. If 2048000 drops of water fill the glass completely then find the height of the glass.

Answer 1

Hint: In this problem, first we need to find the volume of spherical drops and conical glass, and hence find the height of the conical glass by comparing the volume of 2048000 spherical drops with the volume of the conical glass.

Complete step by step solution:
The volume V of a drop of water is obtain as follows:
\[
  \,\,\,\,\,\,V = \dfrac{4}{3}\pi {r^3} \\
   \Rightarrow V = \dfrac{{\pi {d^3}}}{6}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {r = \dfrac{d}{2}} \right) \\
   \Rightarrow V = \dfrac{{\pi {{\left( {\dfrac{1}{{10}}} \right)}^3}}}{6}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {d = \dfrac{1}{{10}}cm} \right) \\
   \Rightarrow V = \dfrac{\pi }{{6000}} \\
\]
Now, consider h be the height of the conical glass and r be the radius of the base of the glass. The volume \[{V_1}\] of the conical glass is calculated as follows:
\[
  \,\,\,\,\,\,{V_1} = \dfrac{1}{3}\pi {r^2}h \\
   \Rightarrow {V_1} = \dfrac{1}{3}\pi {\left( {\dfrac{h}{2}} \right)^2}h \\
   \Rightarrow {V_1} = \dfrac{\pi }{{12}}{h^3} \\
\]
According to the question,
\[
  \,\,\,\,\,{\text{Volume of 2048000 spherical drops}} = {\text{volume of glass}} \\
   \Rightarrow {\text{2048000}}\left( V \right) = {V_1} \\
   \Rightarrow {\text{2048000}}\left( {\dfrac{\pi }{{6000}}} \right) = \dfrac{\pi }{{12}}{h^3} \\
   \Rightarrow {\text{2048}}\left( {\dfrac{\pi }{6}} \right) = \dfrac{\pi }{{12}}{h^3} \\
   \Rightarrow {h^3} = 4096 \\
   \Rightarrow h = 16cm \\
\]

Thus, the height of the conical glass is 16 cm.
Note: The volume of 2048000 spherical droplets is equal to the volume of the conical glass. The formula for the volume of the sphere in terms of diameter is \[V = \dfrac{{\pi {d^3}}}{6}\], here d is diameter of the sphere.