Question

# The length, the breadth and the height of a cuboid are in the ratio of 5:3:2. If it’s volume is $240c{{m}^{3}}$, then find the total surface area of the cuboid in $c{{m}^{3}}$.

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Hint: We will use the formula for the volume of the cuboid to solve this question. If l is the length of a cuboid, b is the breadth of a cuboid and h is the height of a cuboid, then the volume of a cuboid is given by $lbh$ and the total surface area is given by $2\left( lb+bh+hl \right).$

It is given in the question that the length, breadth and height of a cuboid are in the ratio of 5:3:2. Let us consider a variable $x$ such that we can write the given ratios in proportions. The length of a cuboid can be written as $5x$, the breadth of cuboid as $3x$ and height of the cuboid as $2x$.
Now, we use the formula for the volume of the cuboid, as the volume of the cuboid is already given in the question. Using this, we can find the value of x and then we can find the length, breadth and height. Then, we can find the total surface area of the cuboid.
All units are in $c{{m}^{3}}$, so we don’t have to do unit conversions.
The volume of the cuboid is given by $lbh$ where $l=5x$, $b=3x$and $h=2x$. Therefore, substituting the values, we get
Volume = $(5x)(3x)(2x)=30{{x}^{3}}$
Now, according to the question, the volume of a cuboid = $240c{{m}^{3}}$. So, we can equate obtained volume to given volume as,
$30{{x}^{3}}=240c{{m}^{3}}$
\begin{align} & \Rightarrow {{x}^{3}}=8c{{m}^{3}} \\ & \Rightarrow x=2cm \\ \end{align}
So, now we can obtain the length of cuboid as $=5\times 2=10cm$, breadth of cuboid $=3\times 2=6cm$ and height of the cuboid $=2\times 2=4cm$.
Now we will find the total surface area of cuboid give by the formula,
\begin{align} & \Rightarrow 2(lb+bh+hl) \\ & \Rightarrow 2(10\times 6+6\times 4+4\times 10) \\ & \Rightarrow 2(60+24+40) \\ & \Rightarrow 2(124) \\ & =248c{{m}^{3}} \\ \end{align}
So, the total surface area of a cuboid is $=248c{{m}^{3}}$.

Note: All units are in $c{{m}^{3}}$, so there is no need of doing unit conversions. The student must remember the important formula of volume and total surface area of the cuboid. When ratios are given, we need to consider a variable $x$ and then we can write ratios in terms of that variable.