Question

A sphere of 10.5 cm radius is melted and cast into a cuboid of maximum volume. The total surface area of such cuboid is approximately:

Answer 1

"volume of sphere = (4/3)* pi * r^3

= (4/3) * pi * (10.5) ^ 3

on solving it comes 11 * 21 * 21

let l = 11, b = 21, h = 21

so S = 2 (lb + bh + hl)

=2 (11 * 21 + 21 * 21 + 21 * 11)

=2(231+441+231)

=2*903

=1806

"

Answer 2

Find the volume of the sphere:

Volume = 4/3 πr³

Volume = 4/3 π (10.5)³ = 4851 cm³

Find the prime factors of 4851:

4851 = 3² x 7² x 11

To get the max surface area:

*Assuming the lengths must be integer

Length = 3 cm (allocate the smallest possible factor)

Breadth = 3 cm (allocate the next smallest possible factor)

Height = 11 x 7 x 7 = 539 cm (The remaining factors)

Total surface area of the cuboid:

Total Surface Area = 2 (3 x 3 + 3 x 539 + 3 x  539)

Total Surface Area = 6486 cm²

Answer: The total surface area is 6486 cm²