A cone of maximum size is carved out from a cube of edge 14 cm. Find the surface area of the remaining solid after the cone is carved out.

We have

Given: length of the edge of cube = 14 cm.

A cone of maximum size is carved out from a cube.

Surface Area of the remaining solid is given by,

Surface Area of the remaining solid = Area of the cube + Curved Surface Area of the cone – Base area of the cone …(i)

First, let’s find area of cube. It is given by,

Area of cube = 6 (side)²

= 6 (14)² [∵ side = 14 cm, as given]

= 6 × 196

= 1176 cm² …(ii)

Now, we need to find curved surface area of cone.

For curved surface area of cone, we have

Radius of cone (r) = 7 cm [∵ diameter of cone = length of edge of the cube ⇒ radius of cone = 1/2 × diameter of cone]

Height of cone (h) = 14 cm [∵ height of cone = length of edge of cube = 14 cm]

Slant length of cone (l) = ?

Slant length of cone is given by √(r² + h²)

⇒ l = √(r² + h²)

⇒ l = √(7² + 14²)

⇒ l = √(49 + 196)

⇒ l = √245

⇒ l = 7√5

Slant length of cone (l) = 7√5 cm

Curved surface area of cone = πrl

= 22/7 × 7 × 7√5

= 154√5 cm² …(iii)

Now, we need to find area of base of the cone.

The base is a solid circle, we have

Radius of base (R) = 7 cm [∵ radius of base = radius of cone = 7 cm]

Area of base of cone = Area of circle = πR²

⇒ Area of base of cone = 22/7 × 7²

= 154 cm² …(iv)

Substituting values of equations (ii), (iii) & (iv) in equation (i), we get

Surface Area of the remaining solid = 1176 + 154√5 – 154 = 1022 + 154√5

Hence, Surface Area of remaining solid = (1022 + 154√5) cm²