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

Given:
A cube of edge 14 cm.
A cone of maximum size is carved out from the cube.

Step 1: Dimensions of the Cone
Height of the cone (h) = 14 cm (same as the cube’s height)
Diameter of the cone’s base = 14 cm ⇒ Radius (r) = 7 cm

Step 2: Surface Area of the Cone
Total surface area (TSA) of a cone = πr(l + r)
Where: r = 7 cm  &  h = 14 cm
Then Slant height (l) = √(r² + h²) = √(49 + 196) = √245 = 7√5 = 15.652

Now,
Total surface area (TSA) of a cone = π × 7 × (7 + 7√5) = 49π(1 + √5)
Using π = 3.1416 and √5 =  2.236:

TSA   =  49 × 3.1416 × (1 + 2.236)
TSA   =  49 × 3.1416 × 3.236
TSA   =  498.1 cm²

Step 3: Surface Area of the Remaining Solid
Surface area of the cube = 6 × (14)² = 6 × 196 = 1176 cm²
Base area of cone = πr² = π × 49 = 153.94 cm²
Curved surface area of cone = πrl = π × 7 × 7√5 = 344.2 cm²

Now,
Remaining surface area = Surface area of cube - base area of cone + curved surface area of cone
Remaining surface area = 1176 - 153.94 + 344.2
Remaining surface area = 1366.26 cm²

Hence:
Surface area of the cone = 498.1 cm²
Surface area of the remaining solid = 1366.26 cm²