Question

A hemispherical bowl of internal diameter 36 cm contains a liquid. This liquid is to be filled in cylindrical bottles of radius 3 cm and height 6 cm. How many bottles are required to empty the bowl?

Answer 1

Hint: Find the volume of the hemispherical bowl and the volume of cylindrical bottles and then divide them to find the numbers of bottles required
Number of bottles = Volume of the hemispherical bowl/Volume of cylindrical bottles.
The volume of a 3-dimensional shape determines the capacity it can hold or the capacity it has, and in the case of a cylinder, it determines the capacity of the cylinder.
The volume of the cylinder is given as \[V = \pi {r^2}h\] where, (r) is the radius of the base.
The volume hemisphere determines the capacity of the hemisphere. The volume of the hemisphere is given as \[V = \dfrac{2}{3}\pi {r^3}\] where, (r) is the radius of the base.

Complete step-by-step answer:

Given the internal diameter of the hemispherical bowl \[d = 36cm\], so its radius will be \[r = \dfrac{{36}}{2} = 18cm\]
Now find the total volume of liquid in the container by using formula \[{V_H} = \dfrac{2}{3}\pi {r^3}\]
Substituting the values in the formula we get,
\[
  {V_H} = \dfrac{2}{3}\pi {r^3} \\
   = \dfrac{2}{3} \times \dfrac{{22}}{7} \times {\left( {18} \right)^3} \\
 \]
Hence by solving we get
\[
  {V_H} = \dfrac{2}{3} \times \dfrac{{22}}{7} \times {\left( {18} \right)^3} \\
   = \dfrac{2}{3} \times \dfrac{{22}}{7} \times 18 \times 18 \times 18 \\
   = 12219.4c{m^3} \\
 \]
Also, the radius of the bottle is given as \[r = 3cm\] and height \[h = 6cm\]
Hence the volume of the bottle can be determined by the volume of cylinder formula given as \[V = \pi {r^2}h\]
Now substitute the values and solve for volume, we get
\[
  V = \pi {r^2}h \\
   = \pi \times {\left( 3 \right)^2} \times 6 \\
   = \dfrac{{22}}{7} \times 9 \times 6 \\
   = 169.7c{m^3} \\
 \]
So the numbers of bottle required to empty the bowl will be equal to
Number of bottles = Volume of the hemispherical bowl /Volume of cylindrical bottles
\[
  N = \dfrac{{12219.4}}{{169.7}} \\
   = 72.005 \\
   \simeq 72 \\
 \]
Hence the numbers of bottle required to empty the bowl \[ = 72\]

Note: To find the capacity of a closed curve/body, the volume is calculated for a three-dimensional object, and the area is calculated for a two-dimensional figure. It is interesting to note here that, every three-dimensional body is originated by rotating/revolving the two-dimensional body.