The curved surface area of a solid metallic right circular cylinder is 1792π cm² and its base radius is 32 cm. It is melted and recast into 42 solid spherical balls of equal radius. What is the surface area (in cm²) of a ball?

Given:

Curved surface area (CSA) of a solid metallic right circular cylinder = 1792π cm²

Base radius of the cylinder (r_c) = 32 cm

The cylinder is melted and recast into 42 solid spherical balls of equal radius.

Formula used:

Curved surface area of a cylinder = 2πr_ch_c (where r_c is the base radius and h_c is the height of the cylinder)

Volume of a cylinder = πr_c²h_c

Volume of a sphere = (4/3)πr_s³ (where r_s is the radius of the sphere)

Surface area of a sphere = 4πr_s²

Calculation:

Height (h_c) of the cylinder using its curved surface area:

CSA = 2πr_ch_c

1792π = 2π × 32 × h_c

1792 = 2 × 32 × h_c

1792 = 64 × h_c

h_c = 1792 / 64

h_c = 28 cm

Volume of cylinder = πr_c²h_c

= π × (32)² × 28

= π × 1024 × 28

= 28672π cm³

Total volume of 42 balls = 42 × Volume of one spherical ball

28672π = 42 × (4/3)πr_s³

28672 = 42 × (4/3)r_s³

28672 = (168/3)r_s³

28672 = 56r_s³

r_s³ = 28672 / 56

r_s³ = 512

r_s = ∛512

r_s = 8 cm

Surface area of a ball = 4πr_s²

= 4π × (8)²

= 4π × 64

= 256π cm²

∴ The correct answer is option 1.