A box opened at the top is made of wood of thickness 3 cm. Its external…

For a box that is open on one face and made of material of uniform thickness, the inner surface area is found by first working out the inner dimensions: any side bounded by two opposite walls loses twice the wall thickness, while a side bounded by only one wall (the closed face) loses the thickness only once. The required inner surface is then the sum of the areas of every interior face that is actually enclosed, excluding the open face.

1. Convert every measurement to the same unit (metres): thickness = 3 cm = 0.03 m, and height = 83 cm = 0.83 m.

2. Since the box is open at the top, the length and breadth are each bounded by two opposite walls, so each loses twice the thickness; the height is bounded only by the closed base, so it loses the thickness only once: Inner length = 1.48 − 2(0.03) = 1.42 m, Inner breadth = 1.16 − 2(0.03) = 1.10 m, Inner height = 0.83 − 0.03 = 0.80 m.

3. The inner surface to be painted is the base plus the four inner walls (there is no top, since the box is open): Inner surface area = (Inner length × Inner breadth) + 2(Inner length × Inner height) + 2(Inner breadth × Inner height) = (1.42 × 1.10) + 2(1.42 × 0.80) + 2(1.10 × 0.80) = 1.562 + 2.272 + 1.76 = 5.594 m².

4. Cost of painting = Inner surface area × rate = 5.594 × ₹150 = ₹839.10.

As an independent check, use the compact open-box formula: inner surface area = (inner base area) + (inner lateral surface area) = lb + 2h(l + b) = (1.42 × 1.10) + 2(0.80)(1.42 + 1.10) = 1.562 + 2(0.80)(2.52) = 1.562 + 4.032 = 5.594 m², the same value as before.

So the cost of painting the box's inner surface is ₹839.10.