Question

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

Answer 1

Height of the cylinder (h)=10 cm

Radius of base of cylinder (r)=4.2 cm

Now,

Volume of cylinder=$\pi r^{2}h$

$=\frac{22}{7}\times 4.2\times 4.2\times 10$

$=554.4c{m}^3$

Volume of hemisphere=$\frac{2}{3}\pi r^3$

$=\frac{2}{3}\times \frac{22}{7}\times 4.2\times 4.2\times 4.2$

$=155.232c{m}^3$

Volume of the rest of the cylinder after scooping out the hemisphere from each end=Volume of cylinder-$2\times$ Volume of hemisphere

$=554.4-2\times 155.232$

$=554.4-310.464$

$=243.936c{m}^3$

The remaining cylinder is melted and converted into a new cylindrical wire of 1.4 cm thickness.

So, radius of cylindrical wire $=0.7cm$

Volume of remaining cylinder=Volume of new cylindrical wire

$243.936=\pi R^{2}H$

$243.936=\frac{22}{7}\times 0.7\times 0.7\times H\Rightarrow H=158.4cm$