From a cube of edge 14 cm, a cone of maximum size is carved out. Find the volume of the cone and of the remaining material, each correct to one place of decimal.

Edge of a cube = 14 cm

Volume = side³ = 14³ = 2744 cm³.

Cone of maximum size is carved out as shown in figure,

Diameter of the cone cut out from it = 14 cm

Radius, r = $\dfrac{\text{diameter}}{2} = \dfrac{14}{2}$ = 7 cm

Height, h = 14 cm

Volume of cone = $\dfrac{1}{3}$ πr²h

$= \dfrac{1}{3} \times \dfrac{22}{7} \times 7^2 \times 14 \\[1em] = \dfrac{1}{3} \times 22 \times 49 \times 2 \\[1em] = \dfrac{2156}{3} \\[1em] = 718.67 \text{ cm}^3.$

Rounding off to one decimal place = 718.8 cm³

Volume of the remaining material = Volume of the cube - Volume of the cone

= 2744 - 718.67

= 2025.33 cm³

Rounding off to one decimal place = 2025.3 cm³

Hence, the volume of the cone is 718.8 cm³ and of the remaining material is 2025.3 cm³.