A largest possible sphere is carved out from a solid wooden cube of side 7 cm. Find :

(i) the volume of sphere

(ii) the percentage of wood wasted in the process.

(Take π = $\dfrac{22}{7}$)

(i) Largest sphere can have diameter = side of cube = 7 cm.

Radius of sphere (r) = $\dfrac{\text{Diameter}}{2} = \dfrac{7}{2}$ = 3.5 cm

By formula,

$\text{Volume of sphere} = \dfrac{4}{3}πr^3 \\[1em] = \dfrac{4}{3} \times \dfrac{22}{7} \times (3.5)^3 \\[1em] = \dfrac{4}{3} \times 22 \times 0.5 \times (3.5)^2 \\[1em] = \dfrac{539}{3} \\[1em] = 179\dfrac{2}{3} \text{ cm}^3.$

Hence, volume of sphere = $179\dfrac{2}{3}$ cm³.

(ii) By formula,

Volume of cube = (side)³

= 7³

= 343 cm³.

Volume of wood left = Volume of cube - Volume of sphere

$= 343 - 179\dfrac{2}{3} \\[1em] = 343 - \dfrac{539}{3} \\[1em] = \dfrac{1029 - 539}{3} \\[1em] = \dfrac{490}{3} \text{ cm}^3.$

Percentage of wood wasted = $\dfrac{\text{Volume of wood left}}{\text{Volume of wood}} \times 100$

Solving,

$\Rightarrow \dfrac{\dfrac{490}{3}}{343} \times 100 \\[1em] \Rightarrow \dfrac{490}{1029} \times 100 \\[1em] \Rightarrow \dfrac{49000}{1029} \\[1em] \Rightarrow \dfrac{1000}{21} \\[1em] \Rightarrow 47\dfrac{13}{21} \%.$

Hence, percentage of wood wasted = $47\dfrac{13}{21}$ %.