### Theory:

1. A metallic cylinder of height $$36 \ cm$$ and radius of base $$8 \ cm$$ is melted and reshapes into a form of a sphere. Find the diameter of the sphere.

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

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

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

$$r^3$$ $$=$$ $$1728$$

$$r = 12$$

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

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 $$=$$ $$33 \ cm$$

Breadth of the rectangular lead piece $$=$$ $$21 \ cm$$

Height of the rectangular lead piece $$=$$ $$14 \ cm$$

Volume of the rectangular lead piece $$=$$ $$lbh$$ cu. units

$$= 33 \times 21 \times 14$$

$$= 9702 \ cm^3$$

$$\text{Number of spherical lead shots} = \frac{\text{Volume of rectangular lead piece}}{\text{Volume of spherical lead shots}}$$

$$=$$ $\frac{9702}{\left(\frac{539}{3}\right)}$

$$=$$ $\frac{9702×3}{539}$

$$=$$ $$54$$

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