Question.
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?
(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?
Solution:
(i) Edge of cube $=10 \mathrm{~cm}$
Length $(l)$ of box $=12.5 \mathrm{~cm}$
Breadth $(b)$ of box $=10 \mathrm{~cm}$
Height $(h)$ of box $=8 \mathrm{~cm}$
Lateral surface area of cubical box $=4(\text { edge })^{2}$
$=4(10 \mathrm{~cm})^{2}$
$=400 \mathrm{~cm}^{2}$
Lateral surface area of cuboidal box $=2[/ h+b h]$
$=[2(12.5 \times 8+10 \times 8)] \mathrm{cm}^{2}$
$=(2 \times 180) \mathrm{cm}^{2}$
$=360 \mathrm{~cm}^{2}$
Clearly, the lateral surface area of the cubical box is greater than the lateral surface area of the cuboidal box.
Lateral surface area of cubical box - Lateral surface area of cuboidal box $=400 \mathrm{~cm}^{2}-360 \mathrm{~cm}^{2}=40 \mathrm{~cm}^{2}$
Therefore, the lateral surface area of the cubical box is greater than the lateral surface area of the cuboidal box by $40 \mathrm{~cm}^{2}$.
(ii) Total surface area of cubical box $=6(\text { edge })^{2}=6(10 \mathrm{~cm})^{2}=600 \mathrm{~cm}^{2}$
Total surface area of cuboidal box
$=2[/ h+b h+l b]$
$=\left[2(12.5 \times 8+10 \times 8+12.5 \times 10] \mathrm{cm}^{2}\right.$
$=610 \mathrm{~cm}^{2}$
Clearly, the total surface area of the cubical box is smaller than that of the cuboidal box.
Total surface area of cuboidal box $-$ Total surface area of cubical box $=610 \mathrm{~cm}^{2}-600 \mathrm{~cm}^{2}=10 \mathrm{~cm}^{2}$
Therefore, the total surface area of the cubical box is smaller than that of the cuboidal box by $10 \mathrm{~cm}^{2}$.
(i) Edge of cube $=10 \mathrm{~cm}$
Length $(l)$ of box $=12.5 \mathrm{~cm}$
Breadth $(b)$ of box $=10 \mathrm{~cm}$
Height $(h)$ of box $=8 \mathrm{~cm}$
Lateral surface area of cubical box $=4(\text { edge })^{2}$
$=4(10 \mathrm{~cm})^{2}$
$=400 \mathrm{~cm}^{2}$
Lateral surface area of cuboidal box $=2[/ h+b h]$
$=[2(12.5 \times 8+10 \times 8)] \mathrm{cm}^{2}$
$=(2 \times 180) \mathrm{cm}^{2}$
$=360 \mathrm{~cm}^{2}$
Clearly, the lateral surface area of the cubical box is greater than the lateral surface area of the cuboidal box.
Lateral surface area of cubical box - Lateral surface area of cuboidal box $=400 \mathrm{~cm}^{2}-360 \mathrm{~cm}^{2}=40 \mathrm{~cm}^{2}$
Therefore, the lateral surface area of the cubical box is greater than the lateral surface area of the cuboidal box by $40 \mathrm{~cm}^{2}$.
(ii) Total surface area of cubical box $=6(\text { edge })^{2}=6(10 \mathrm{~cm})^{2}=600 \mathrm{~cm}^{2}$
Total surface area of cuboidal box
$=2[/ h+b h+l b]$
$=\left[2(12.5 \times 8+10 \times 8+12.5 \times 10] \mathrm{cm}^{2}\right.$
$=610 \mathrm{~cm}^{2}$
Clearly, the total surface area of the cubical box is smaller than that of the cuboidal box.
Total surface area of cuboidal box $-$ Total surface area of cubical box $=610 \mathrm{~cm}^{2}-600 \mathrm{~cm}^{2}=10 \mathrm{~cm}^{2}$
Therefore, the total surface area of the cubical box is smaller than that of the cuboidal box by $10 \mathrm{~cm}^{2}$.