Question

6 tennis balls of diameter 62mm are placed in a cylindrical tube as shown in the figure. Find the volume of the internal unfilled space in the tube and express this as the percentage of the volume of the tube.

Answer 1

Hint: We will find the volume of the tennis balls and the volume of the tube. We will subtract the volume of the tennis balls from the volume of the tube. Then, we will express the volume as the percentage of the volume of the tube by using the formula for percentage.

Formulas used:
1. Volume of a cylinder is given by \[{V_C} = \pi {r^2}h\] where \[r\] is the radius and \[h\] is the height of the cylinder.
2. Volume of a sphere is given by \[{V_S} = \dfrac{4}{3}\pi {r^3}\] where \[r\] is the radius of the sphere.
3. Formula of Percentage is \[\dfrac{{{\rm{given\ value}}}}{{{\rm{total\ value}}}} \times 100\]
4. \[{\rm{Radius}} = \dfrac{{{\rm{Diameter}}}}{2}\]

Complete step-by-step answer:
Each tennis ball is a sphere and the tube is a cylinder.

We will calculate the radius of the sphere by substituting 62 for diameter in the formula \[{\rm{Radius}} = \dfrac{{{\rm{Diameter}}}}{2}\]:
\[\begin{array}{l}r = \dfrac{{62}}{2}\\ \Rightarrow r = 31{\rm{mm}}\end{array}\]
The radius of the sphere is 31 mm. We will calculate the volume of the sphere by substituting 31 for \[r\] in the formula, \[{V_S} = \dfrac{4}{3}\pi {r^3}\]. Therefore, we get
 \[{V_S} = \dfrac{4}{3}\pi {\left( {31} \right)^3}\]
Applying the exponent on the term, we get
\[ \Rightarrow {V_S} = \dfrac{4}{3}\pi \left( {29791} \right)\]
Simplifying the above expression, we get
\[ \Rightarrow {V_S} = 124788.279{\rm{m}}{{\rm{m}}^3}\]
We will calculate the volume occupied by 6 spheres:
 \[6{V_S} = 6 \times 124788.249 = 748729.494{\rm{m}}{{\rm{m}}^3}\]
We will calculate the volume of the cylinder.
We can see in the figure that the radius of the cylinder is the same as the radius of the sphere and the height of the cylinder is 6 times the diameter of the sphere.
\[r = 31{\rm{mm}}\]
\[\begin{array}{l}h = 6 \times 62\\ \Rightarrow h = 372{\rm{mm}}\end{array}\]
We will substitute 31 for \[r\] and 372 for \[h\] in the formula \[{V_C} = \pi {r^2}h\] and calculate the volume of the cylinder:
\[\begin{array}{l}{V_C} = \pi {\left( {31} \right)^2} \cdot 372\\ \Rightarrow {V_C} = 1123094.24{\rm{m}}{{\rm{m}}^3}\end{array}\]
The volume of internal unoccupied space will be the difference of the volume of the cylinder and the volume of 6 spheres.
\[V = {V_C} - 6{V_S}\]
Substituting the values of volumes, we get
\[ \Rightarrow V = 1123094.24 - 748729.494\]
Subtracting the terms, we get
\[ \Rightarrow V = 374364.746{\rm{m}}{{\rm{m}}^3}\]
We will express this volume as a percentage of the volume of the tube. We will substitute 374364.746 for the given volume and 1123094.24 for the total volume in the formula, \[\dfrac{{{\rm{given\ value}}}}{{{\rm{total\ value}}}} \times 100\]. Therefore, we get
\[\dfrac{{374364.746}}{{1123094.24}} \times 100 = 33.33\% \]
$\therefore $ The volume of the internal unfilled space is \[374364.746{\rm{ m}}{{\rm{m}}^3}\] and it is \[33.33\% \] of the total volume of the cylinder.
Note: We might find it difficult to find the height of the tube. We should look at the figure and geometrically determine the height. The height is 6 times the diameter of a tennis ball or 12 times the radius of the tennis ball.