Question

6 Tennis balls diameter 70mm. Each are placed in a cylindrical card tube. Prove that volume of unfilled space in the tube is 33.33% of volume of the tube.

Answer 1

Answer :

Diameter of a tennis balls = 70 mm = 7 cm ( As we know 1 cm = 10 mm )
And
Total tennis balls = 6
So,
Height of Cylindrical card tube = 6 $\times$ 7 = 42 cm

Radius of Cylinder = Radius of tennis ball ( spherical ) = 3.5 cm ( As diameter of balls given as 7 cm )

We know Area of Cylinder = $\pi r^{2}h$
So,
Volume of Cylindrical card tube= $\frac{22}{7}\times 3.5\times 3.5\times 42=22\times 3.5\times 3.5\times 6\mathbf{=}\mathbf{}\mathbf{}\mathbf{1617}\mathbf{}\mathit{c}{\mathit{m}}^{\mathbf{3}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}(Asweknow\pi =\frac{22}{7})$

And
We know Volume of sphere = $\frac{4\pi r^3}{3}$
So,
Volume of a tennis ball = $\frac{4\times {\displaystyle \frac{22}{7}}\times 3.5\times 3.5\times 3.5}{3}=\frac{4\times 22\times 0.5\times 3.5\times 3.5}{3}=\frac{539}{21}\mathbf{=}\mathbf{}\mathbf{}\mathbf{179}\mathbf{.}\mathbf{6666667}\mathit{c}{\mathit{m}}^{\mathbf{3}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}(Asweknow\pi =\frac{22}{7})$
Som
Volume of all 6 tennis balls = 6 $\times$179.6666667 = 1078 cm³

And

Volume of unfilled space = 1617 - 1078 = 539

So,
Percentage of volume of unfilled space in the tube of volume of the tube. = $\frac{539}{1617}\times 100=0.333\times 100\mathbf{=}\mathbf{}\mathbf{}\mathbf{33}\mathbf{.}\mathbf{3}\mathbf{}\mathbf{\%}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{(}\mathbf{}\mathit{H}\mathit{e}\mathit{n}\mathit{c}\mathit{e}\mathbf{}\mathit{p}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{d}\mathbf{}\mathbf{)}$