A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm long, 10 cm wide and 8 cm high.
i) Which box has the greater lateral surface area and by how much?
ii) Which box has the smaller total surface area and by how much?

Given:
Edge of cube = 10 cm
Length (l) of box = 12.5 cm
Breadth (b) of box = 10 cm
Height (h) of box = 8 cm

i) We know that, Lateral surface area of cubical box
= \(4 × (Edge)^2\)
= \(4 × (10)^2 {cm}^2\)
= 400 \({cm}^2\)

Also, We have,

Lateral surface area of cuboidal box
= 2 [lh + bh]
= [2 (12.5 x 10 x 8)] \({cm}^2\)
= [2 x 180] \({cm}^2\)
= 360 \({cm}^2\)

Clearly, the lateral surface area of the cubical box is greater than the lateral surface area of the cuboidal box.

Now, Lateral surface area of cubical box - Lateral surface area of cuboidal box
= 400 \({cm}^2\) - 360 \({cm}^2\) = 40 \({cm}^2\)

Therefore, the lateral surface area of the cubical box is greater than the lateral surface area of the cuboidal box by 40 \({cm}^2\)


ii)Similarly, Total surface area of cubical box
=\(6 × (Edge)^2\)
= \(6 × (10)^2 {cm}^2\)
= 600 \({cm}^2\)

Also, We have,

Total surface area of cuboidal box
= 2(lh + bh + lb)
=[2 (12.5 x 8 + 10 x 8 + 12.5 x 100)] \({cm}^2\)
= 610 \({cm}^2\)

Clearly, the total surface area of the cubical box is smaller than that of the cuboidal box.

Now, Total surface area of cuboidal box - Total surface area of cubical box
= 610 \({cm}^2\) - 600\({cm}^2\)
= 10 \({cm}^2\)

Therefore, the total surface area of the cubical box is smaller than that of the cuboidal box by 10 \({cm}^2\).