Question

Find the order of magnitude of the number of table tennis balls that would fit into a typical-size room (without being crushed).

Answer 1

Hint: Use the formula for volume of a spherical object in terms of diameter of the object. Use the approximate value of diameter of the tennis ball and typical size of a room. Calculate the volume of the table tennis ball and the room. Then divide the volume of the room by the volume of a table tennis ball to determine the number of ball that can fit into the room.

Formula used:
The volume \[V\] of a spherical object is given by
\[V = \dfrac{{\pi {d^3}}}{6}\] …… (1)
Here, \[d\] is the diameter of the spherical object.

Complete step by step answer:
We have asked to determine the order of magnitude of the number of table tennis balls that would fit into a typical size room.For this, we should know the volume of a table tennis ball and volume of the typical size room.We know that the diameter of a table tennis ball is \[6.54\,{\text{cm}}\].
\[d = 6.54\,{\text{cm}}\]
Let us first determine the volume of a table tennis ball.Substitute \[3.14\] for \[\pi \] and \[6.54\,{\text{cm}}\] for \[d\] in equation (1).
\[{V_T} = \dfrac{{\left( {3.14} \right){{\left( {6.54\,{\text{cm}}} \right)}^3}}}{6}\]
\[ \Rightarrow {V_T} = 146.4\,{\text{c}}{{\text{m}}^3}\]
Hence, the volume of a table tennis ball is \[146.4\,{\text{c}}{{\text{m}}^3}\].

The typical size room has dimensions \[\left( {15 \times 20 \times 8} \right)\,{\text{f}}{{\text{t}}^3}\].Let us now calculate the volume of the typical size room.
\[V = \left( {15 \times 20 \times 8} \right)\,{\text{f}}{{\text{t}}^3}\]
\[ \Rightarrow V = 2400\,{\text{f}}{{\text{t}}^3}\]
\[ \Rightarrow V = \left( {2400\,{\text{f}}{{\text{t}}^3}} \right)\left( {\dfrac{{{{30}^3}\,{\text{c}}{{\text{m}}^3}}}{{1\,{\text{f}}{{\text{t}}^3}}}} \right)\]
\[ \Rightarrow V = 64800000\,{\text{c}}{{\text{m}}^3}\]
Hence, the volume of the typical size room is \[64800000\,{\text{c}}{{\text{m}}^3}\].
Now we can calculate the number of the table tennis balls that can fit into the room.The number of table tennis balls that can fit into the room are
\[N = \dfrac{V}{{{V_T}}}\]
Substitute \[64800000\,{\text{c}}{{\text{m}}^3}\] for \[V\] and \[146.4\,{\text{c}}{{\text{m}}^3}\] for \[{V_T}\] in the above equation.
\[N = \dfrac{{64800000\,{\text{c}}{{\text{m}}^3}}}{{146.4\,{\text{c}}{{\text{m}}^3}}}\]
\[ \Rightarrow N = 44262.3\]
\[ \therefore N = 4.4 \times {10^4}\]
Therefore, the number of table tennis balls that can fit into the room are \[4.4 \times {10^4}\].

Hence, the order of magnitude of the number of table tennis balls that would fit into a typical size room is \[{10^4}\].

Note: The students should keep in mind that we have calculated the number of balls that can fit in the room by considering that the total volume of the room can be filled with the balls. But in reality this is not possible as the packing fraction of the most ideal FCC lattice structure is also not 1 but 0.74. So, in reality, this number that we have calculated is more than the actual number of balls that can be fitted.