From each end of a solid metal cylinder, metal was scooped out in the hemispherical form of the same diameter. The height of the cylinder is 10 cm and its base is of radius 4.2 cm. The rest of the cylinder is melted and converted into a cylindrical wire of 1.4 cm thickness. Find the length of the wire. ( π = 22/7)

Solution

Height of the solid metal cylinder, h = 10 cm
Radius of the solid metal cylinder, r = 4.2 cm
therefore,
Radius of each hemisphere = radius of the solid metal cylinder, r = 4.2 cm
Now,
Volume of the rest of the cylinder = Volume of cylinder - 2 x volume of each hemisphere
$space equals space πr squared straight h space minus space 2 space straight x space 2 over 3 space πr cubed space equals space πr squared space open parentheses straight h space minus 4 over 3 straight r close parentheses equals straight pi space straight x space left parenthesis 4.2 space right parenthesis squared space open parentheses 10 space minus space 4 over 3 space straight x space 4.2 close parentheses space equals straight pi space straight x space left parenthesis 4.2 right parenthesis squared space straight x space left parenthesis 4.4 right parenthesis space cm cubed$
Thickness of the cylindrical wire = 1.4 cm
Therefore,
Radius of the cylindrical wire, R = 1.4/2 = 0.7 cm
Let the length of the wire be H cm.
It is given that the rest of the cylinder is melted and converted into a cylindrical wire.
therefore,
Volume of the cylindrical wire = Volume of the rest of the cylinder
⇒ π x 0.7 x 0.7 x H = π x (4.2)² x (4.4)
$rightwards double arrow space straight H space equals space fraction numerator 4.2 space straight x space 4.2 space straight x space 4.4 over denominator 0.7 space straight x space 0.7 end fraction space equals space 158.4 space cm$
Hence, the length of the wire is 158.4 cm

Some More Questions From Statistics Chapter

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

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