The volume of a cubical solid is 10368 cm³. If its dimensions are in the ratio 3 : 2 : 1, find the cost of polishing its total surface at the rate of ₹ 2.50 per m².

Volume of cuboid = 10368 cm³

Ratio of dimensions = 3 : 2 : 1

Let the dimensions be 3k, 2k, 1k.

Volume of cuboid = l x b x h

⇒ 10368 = 3k x 2k x k

⇒ 10368 = 6k³

⇒ k³ = $\dfrac{10368}{6}$

⇒ k³ = 1728

⇒ k = $\sqrt[3]{1728}$

⇒ k = 12

So, the dimensions are:

Length = 3k = 3 x 12 = 36 cm

Breadth = 2k = 2 x 12 = 24 cm

Height = k = 12 cm

The total surface area of a cuboid = 2(lb + bh + hl)

= 2(36 x 24 + 24 x 12 + 12 x 36)

= 2(864 + 288 + 432)

= 2 x 1584

= 3168 cm²

= 0.3168 m²

Rate of polishing = ₹ 2.50 per m²

Total cost = Surface area x Rate

= 0.3168 x 2.50

= ₹ 0.7920

Hence, the cost of polishing the total surface area = ₹ 0.7920.