Question

From one face of a solid cube of side 14 cm, the largest possible one is carved out. Find the volume and surface area of the remaining solid. (Use π = `22/7`, `sqrt5 = 2.2`)

Answer 1

Given:

A cube of side 14 cm

The largest possible cylinder is carved out from one face of the cube.

The diameter of the largest possible cylinder is equal to the side of the cube, i.e., 14 cm.

So, radius rrr of the cylinder r = `14/2`

= 7 cm

The height h of the cylinder is equal to the side of the cube, i.e., 14 cm.

The volume of a cylinder is given by:

V = πr²h

Substituting values,

V = π(7)²(14)

= π(49)(14)

= 686π cm³

Approximating π ≈ 3.1416:

V ≈ 686 × 3.1416 = 2155.13 cm³

The volume of the cube:

V_cube = side³

= 14³

= 2744 cm³

Volume of the remaining solid:

`V_"remaining"​ = V_"cube"​ − V_"cylinder​"`

= 2744 − 2155.13

= 588.87 cm³

Surface Area of Cube (Before Carving Out Cylinder)

A_cube ​= 6s²

= 6(14)²

= 6(196)

= 1176 cm²

Total surface area of a closed cylinder:

`A_"cylinder" = 2πr^2 + 2πrh`

= 2π(7)² + 2π(7)(14)

=2π(49) + 2π(98)

= 98π + 196π

= 294π

Approximating π ≈ 3.1416:

`A_"cylinder"` ≈ 294 × 3.1416

= 923.58 cm²

`A_"remaining​" = A_"cube"`​− Base of Cylinder + Curved Surface of Cylinder

= 1176 − π(7²) + 2π(7)(14)

= 1176 − 49π + 196π

= 1176 + 147π

Approximating π ≈ 3.1416:

`A_"remaining"` = 1176 + (147 × 3.1416)

​= 1176 + (147 × 3.1416)

= 1176 + 461.87

= 1637.87 cm²