Question

The length, width and height of a rectangular solid are in the ratio of $3:2:1$. If the volume of the box is $48c{{m}^{3}}$, the total surface area of the box is
$\left( A \right)\text{ }88c{{m}^{2}}$
$\left( B \right)\text{ 32}c{{m}^{2}}$
$\left( C \right)\text{ 64}c{{m}^{2}}$
$\left( D \right)\text{ }216c{{m}^{2}}$

Answer 1

Hint: In this question we have been given the volume of a rectangular solid and the ratio of the length, width and height are given as $3:2:1$. Given this data, we have to find the total surface area of the box. We will solve this question by considering a common multiple $x$ for the ratios and use the formula $V=l\times b\times h$ and substitute and solve for $x$. Once we find $x$, we will substitute it in the original ratio and solve for the total surface area of the rectangular solid using the formula $SA=2\left( lb+bh+hl \right)$ and get the required solution.

Complete step by step answer:
We have the volume of the solid given to us as $48c{{m}^{3}}$. We have the ratio of the length, breadth and height as $3:2:1$, considering a common multiple $x$, we can write the dimensions as:
$l=3x$
$b=2x$
$h=x$
The rectangular can be represented as:

Now we know that $V=l\times b\times h$ and we know that $V=48c{{m}^{3}}$. On substituting the values, we get:
$\Rightarrow 48=3x\times 2x\times 1x$
On multiplying, we get:
$\Rightarrow 48=6{{x}^{3}}$
On rearranging and simplifying, we get:
$\Rightarrow {{x}^{3}}=8$
Now we know that ${{2}^{3}}=8$ therefore, on substituting, we get:
$\Rightarrow {{x}^{3}}={{2}^{3}}$
Taking the square root on both the sides, we get:
$\Rightarrow x=3$
Therefore, we have the dimensions as:
$l=6$
$b=4$
$h=2$
Now we know that $SA=2\left( lb+bh+hl \right)$ therefore, on substituting the values, we get:
$\Rightarrow SA=2\left( 6\times 4+4\times 2+2\times 6 \right)$
On multiplying the terms, we get:
$\Rightarrow SA=2\left( 24+8+12 \right)$
On adding the values, we get:
$\Rightarrow SA=2\left( 44 \right)$
On multiplying, we get:
$\Rightarrow SA=88c{{m}^{2}}$, which is the required solution.
Therefore, the total surface area of the rectangular solid is $88c{{m}^{2}}$.

So, the correct answer is “Option A”.

Note: It is to be noted that the width of the rectangular solid is the breadth of the solid. Generally, the term breadth is used in the formulas. It is to be remembered that the unit of the volume of a solid and the surface area of the solid is different. The unit for volume is $c{{m}^{3}}$ and for the total surface area is $c{{m}^{2}}$.