The external dimensions of a closed wooden box are 27 cm, 19 cm and 11 cm. If the thickness of the wood in the box is 1.5 cm; find :

(i) volume of the wood in the box;

(ii) the cost of the box, if wood costs ₹ 1.20 per cm³;

(iii) number of 4 cm cubes that could be placed into the box.

(i) Given:

External dimensions of wooden box = 27 cm, 19 cm and 11 cm.

Thickness of the wood = 1.5 cm

External volume of box = l x b x h

= 27 x 19 x 11

= 5,643 cm ³

Internal volume of box = l x b x h

= (27 - 2 x 1.5) x (19 - 2 x 1.5) x (11 - 2 x 1.5)

= (27 - 3) x (19 - 3) x (11 - 3)

= 24 x 16 x 8

= 3,072 cm ³

Volume of wood = External volume of box - Internal volume of box

= 5,643 cm ³ - 3,072 cm ³

= 2,571 cm ³

Hence, volume of wood in the box is 2,571 cm ³.

(ii) Cost of wood = ₹ 1.20 per cm³

Total cost = Volume of wood x Cost of wood

= ₹ 2,571 x 1.20

= ₹ 3,085.20

Hence, total cost of the box is ₹ 3,085.20.

(iii) Side of cube = 4 cm

Let n be the number of cubes.

Internal volume of box = Number of cubes x Volume of cube

⇒ 3,072 = n x side³

⇒ 3,072 = n x 4³

⇒ 3,072 = n x 64

⇒ n = $\dfrac{3,072}{64}$

⇒ n = 48

Hence, the number of 4 cm cubes that could be placed into the box is 48.