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Mensuration — MATHS STD 8 — Question

Gujarat BoardEnglish MediumSTD 8MATHSMensuration5 Marks
Question
A company sells biscuits. For packing purpose they are using cuboidal boxes : Box $A \rightarrow 3 cm \times 8 cm \times 20 cm$ and Box $B \rightarrow 4 cm \times 12 cm \times 10 cm$. What size of the box will be economical for the company? Why? Can you suggest any other size (dimensions) which has the same volume but is more economical than these?
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Answer

For Box A,
Given, length of box $=20 cm$
Breadth of box $=8 cm$ and height of $box =3 cm$
$\therefore$ Volume of the box $=1 \times b \times h=20 \times 8 \times 3=480 cm^{3}$
Total surface area of the box $=2(l b+b h+h t)$
$\begin{array}{l}=2(20 \times 8+8 \times 3+3 \times 20) \\ =2 \times(160+24+60)=2 \times 244 \\ =488 cm^2\end{array}$
For Box B,
Given, length of box $=12 cm$
Breadth of box $=10 cm$ and height of box $=4 cm$
$\therefore$ Volume of the box $=l \times b \times h=12 \times 10 \times 4=480 cm^3$
Total surface area of the box $=2(l b+b h+h l)$
$\begin{array}{l}=2(12 \times 10+10 \times 4+4 \times 12) \\ =2(120+40+48) \\ =2 \times 208=416 cm^2\end{array}$
It is clear from above that both boxes have same volume but the surface area of type $B$ box is less than that of type $A$ box. Thus, material required for Box $B$ is less.
$\therefore$ Box $B$ is more economical than Box $A$.
Now, let another box of size be $12 cm \times 8 cm \times 5 cm$.
Volume of this box $=12 \times 8 \times 5=480 cm^3$
and its surface area $=2(l b+b h+h l)$
$\begin{array}{l}=2(12 \times 8+8 \times 5+5 \times 12) \\ =2(96+40+60)=2 \times 196=392 cm^2\end{array}$
Clearly, surface area of this box is less than that of Box $B$.
Hence, we can suggest any other box of size $12 cm \times 8 cm \times 5 cm$, which has the same volume but is more economical than the given boxes.

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