Some plastic balls of radius 1 cm were melted and cast into a tube. The thickness, length and outer radius of the tube were 2 cm, 90 cm, and 30 cm respectively. How many balls were melted to make the tube?
Solution
The radius of each plastic ball, R = 1 cm
Outer radius of the tube, r2 = 30 cm
The thickness of the tube = 2 cm
∴ The inner radius of the tube, r1 = Outer radius of the tube − Thickness of the tube = 30 − 2 = 28 cm
Length of the tube, h = 90 cm
Let the number of plastic balls melted to make the tube be n.
It given that plastic balls are melted to form the tube.
∴ n × Volume of each plastic ball = Volume of the tube
\[\Rightarrow n = \frac{\text{ Volume of the tube} }{\text{ Volume of each plastic ball } }\]
\[ \Rightarrow n = \frac{\pi\left( r_2^2 - r_1^2 \right)h}{\frac{4}{3}\pi R^3}\]
\[ \Rightarrow n = \frac{\left( {30}^2 - {28}^2 \right) \times 90}{\frac{4}{3} \times 1^3}\]
\[ \Rightarrow n = \frac{116 \times 3 \times 90}{4} = 7830\]
Thus, the number of plastic balls melted to make the tube are 7830.
