2. Problems based on conversion of solid from one shape to another

 

Solution:

 

Height of the cylinder $=$ $36 \ cm$

 

The radius of the base of the cylinder $=$ $8 \ cm$

 

Volume of the cylinder $=$ $\pi r^2 h$ cu. units

 

$=$ $\pi \times 8^2 \times 36$

 

$= 2304 \pi \ cm^3$

 

Let '$r$' be the radius of the sphere.

 

Volume of sphere $=$ $\frac{4}{3} \pi r^3$ cu. units

 

Since the cylinder reshapes into a sphere, the volume remains unchanged.

 

Volume of sphere $=$ Volume of the cylinder

 

$\frac{4}{3} \pi r^3 = 2304 \pi$

 

$r^3$ $=$ $\frac{2304\mathrm{\pi }\times 3}{\mathrm{\pi }\times 4}$

 

$r^3$ $=$ $1728$

 

$r = 12$

 

$d = r \times 2 = 12 \times 2 = 24$

 

Therefore, the diameter of the sphere is $24 \ cm$.

 

 

2. How many spherical lead shots of radius $3.5 \ cm$ can be made out of a solid rectangular lead piece with dimensions $33 \ cm$, $21 \ cm$ and $14 \ cm$.

 

Solution:

 

The radius of the spherical lead shot $=$ $3.5 \ cm$

 

Volume of spherical lead shot $=$ $\frac{4}{3} \pi r^3$ cu. units

 

$=$ $\frac{4}{3} \times \frac{22}{7} \times (3.5)^3$

 

$=$ $\frac{539}{3}$ $cm^3$

 

Length of the rectangular lead piece $=$ $33 \ cm$

 

$=$ $\frac{9702\times 3}{539}$

 

$=$ $54$

 

Therefore, $54$ spherical leads can be made from the rectangular lead piece.