A spherical iron shell with external diameter 21 cm weighs \(22775\fra

1.	A spherical iron shell with external diameter 21 cm weighs \(22775\frac5{21}\) grams. Find the thickness of the shell if the metal weighs 10 gms per cu cm. (a) 3 cm (b) 1 cm (c) 2 cm (d) 2.5 cm
Answer» (c) 2 cm. Let the internal radius of the shell be r cm. ∴ Internal volume of the shell = \(\frac43πr^3\) cu. cm External radius of the shell = \(\frac{21}{2}\) cm. External volume of the shell = \(\frac43\times\frac{22}{7}\times\frac{21}{2}\times\frac{21}{2}\times\frac{21}{2}\) cu. cm = 4851 cu. cm. Weight of 1 cu. cm of metal = 10g. ∴ Volume of the metal in the shell = \(22775\frac5{21}\times \frac{1}{10}\) cu. cm = \(\frac{478280}{21}\times\frac{1}{10}\) cu. cm = \(\frac{47828}{21}\) cu. cm ∴ Internal vol. of the shell = \(4851-\frac{47828}{21}=\frac{54043}{21}\) cu. cm ⇒ \(\frac43πr^3=\frac{54043}{21}\) ⇒ r³= \(\frac{54043\times3\times7}{21\times4\times22}\) = 614.125 ⇒ \(r = 8.5 \) cm ∴ Thickness of shell = 10.5 cm – 8.5 cm = 2 cm

A spherical iron shell with external diameter 21 cm weighs \(22775\frac5{21}\) grams. Find the thickness of the shell if the metal weighs 10 gms per cu cm. (a) 3 cm (b) 1 cm (c) 2 cm (d) 2.5 cm

Answer»

(c) 2 cm.

Let the internal radius of the shell be r cm.

∴ Internal volume of the shell = \(\frac43πr^3\) cu. cm

External radius of the shell = \(\frac{21}{2}\) cm.

External volume of the shell = \(\frac43\times\frac{22}{7}\times\frac{21}{2}\times\frac{21}{2}\times\frac{21}{2}\) cu. cm

= 4851 cu. cm.

Weight of 1 cu. cm of metal = 10g.

∴ Volume of the metal in the shell = \(22775\frac5{21}\times \frac{1}{10}\) cu. cm

= \(\frac{478280}{21}\times\frac{1}{10}\) cu. cm = \(\frac{47828}{21}\) cu. cm

∴ Internal vol. of the shell = \(4851-\frac{47828}{21}=\frac{54043}{21}\) cu. cm

⇒ \(\frac43πr^3=\frac{54043}{21}\) ⇒ r³= \(\frac{54043\times3\times7}{21\times4\times22}\) = 614.125