Question

Four identical cones each of radius 10.5 cm and height 14 cm are cut from a cuboid of dimensions 30 cm × 32 cm × 40 cm (base of each cone lies on the surface of cuboid). What is the total surface area (in cm²) of the remaining solid?
1. 6528
2. 7804
3. 5926
4. 6824

Answer 1

Correct Answer - Option 2 : 7804

Given:

Cuboid has a dimension = 30 cm × 32 cm × 40 cm

Four identical cones each of radius 10.5 cm and height 14 cm are cut from the cuboid

Formulae Used:

Total Surface Area of a cuboid = 2 × [(l × b) + (b × h) + (h × l)],

where l, b, and h = length, breadth, and height of the cuboid; respectively

Curved Surface Area of a Cone = π × r × L,

where r = radius of base, L = slant height of the cone = √(r² + h²), and h = height of the cone

Area of the base of the Cone = π × r²

Calculation:

We need to calculate the area, that will be left over after cutting out the four cones.

Hence, the required surface area contains the area due to the slant heights of the 4 cones, total surface area of the cuboid;                                                                   but without calculating the area of the base of the cone.

The required surface area = Total Surface Area of a cuboid + [4 × (Curved Surface Area of a Cone – Area of the base of the Cone)]      ----(i)

We have:

Total Surface Area of a cuboid = 2 × [(30 × 32) + (32 × 40) + (40 × 30)]

⇒ Total Surface Area of a cuboid = 6880 cm²      ----(ii)

Also, we have:

Curved Surface Area of a Cone = π × r × L = π × r × √(r2 + h2)

⇒ Curved Surface Area of a Cone = (22/7) × 10.5 × √(14² + 10.5²) = 577.5 cm²      ----(iii)

Also, we have:

Area of the base of the Cone = π × r² = (22/7) × 10.5 × 10.5 = 346.5 cm²      ----(iv)

On substituting the values from equations (ii), (iii), and (iv) into equation (i); we get:

The required surface area = 6880 + [4 × (577.5 – 346.5)]   

⇒ The required surface area = 6880 + (4 × 231)   

⇒ The required surface area = 6880 + 924 = 7804   

∴ The required surface area = 7804 cm2