Question

The length, breadth and height of c cuboid are in the ratio 5:3:2. If the volume is $35.937{{m}^{3}}$, find its dimensions. Also find the surface area of the cuboid.

Answer 1

Hint: Assume a proportionality constant (say x) for the given ratio of length, breadth and height. Calculate length, breadth and height in terms of x. then use the formula $''Volume=length\times breadth\times height''$ to find volume of the cuboid in terms of x and equate it with given volume to obtain equation in x and finally solve the equation to get x and using x find dimension of cuboid. Find the total surface area of the cuboid using formula $2\left( lb+bh+lh \right)$ where l=length, b=breadth and h=height.

Complete step-by-step answer:


We have to find the dimension of the cuboid whose ratio of length, breadth and height is given.

Given ratio- length: breadth: height=5:3:2

Let the proportionality constant to be x

So, length of the cuboid will be 5x and

Breadth of the cuboid will be 3x and

Height of the cuboid will be 2x.

We know that volume of the cuboid= $length\times breadth\times height$

By putting length=5x, breadth=3x and height=2x, we will get

Volume of this cuboid= $\left( 5x \right)\left( 3x \right)\left( 2x \right)$

But, according to the question, the volume of this cuboid = $35.937{{m}^{3}}$.

So,
$\begin{align}
  & \left( 5x \right)\left( 3x \right)\left( 2x \right)=35.937{{m}^{3}} \\
 & \Rightarrow 30{{x}^{3}}=35.937{{m}^{3}} \\
\end{align}$

On dividing both sides by 30, we will get-

\[\begin{align}

  & \Rightarrow {{x}^{3}}=\dfrac{35.937}{30} \\

 & \Rightarrow {{x}^{3}}=1.1979 \\

\end{align}\]

On taking cube root of both sides of equation, we will get-

$\begin{align}

  & \Rightarrow x=\sqrt{1,1979} \\

 & \Rightarrow x-1.1062m \\

\end{align}$

So, length$=5x=5\times \left( 1.1062 \right)=5.531m$

     Breadth$=3x=3\times \left( 1.1062 \right)=3.3186m$

     Height$=2x=2\times \left( 1.1062 \right)=2.2124m$

On putting the above obtained value of length, breadth and height in the formula of volume

We will get-

Volume of the cuboid=$\left( 5.531m \right)\left( 3.3186m \right)\left( 2.2124m \right)$

                                        = $40.6089{{m}^{3}}$

Now, let us find out the total surface area of the cuboid.

We know that formula for TSA of cuboid= 2(lb+bh+lh)

Where l= length

                b=breadth

                h=height

$\Rightarrow $TSA of cuboid

\[\begin{align}

  & =2+\left[ \left( 5.531 \right)\left( 3.3186 \right)+\left( .3186 \right)\left( 2.2124 \right)+\left( 5.531 \right)\left( 2.2124 \right) \right] \\

 & =2\left[ 18.355+7.342+12.237 \right] \\

 & =2\left[ 37.334 \right] \\

 & =2\times 37.334 \\

 & =75.862{{m}^{2}} \\

\end{align}\]

Hence, the required volume of the cuboid is $40.6080{{m}^{3}}$ and total surface area of the cuboid is $75.868{{m}^{2}}$.

Note: We have used the formula for TSA of cuboid= 2(lb+bh+lh). We can derive the formula in the following way-
 Total surface area= sum of area of all 6 faces
 = (area of face 1) + (area of face 2) + (area of face 3) + (area of face 4) + (area of face 5) + (area of face 6)
$\Rightarrow $Total surface area

$=\left[ \begin{align}

  & \left( length\,of\,face\,\,1 \right)\times \left( Breadth\,\,of\,face\,1 \right)+\left( length\,of\,face\,2 \right)\times \left( breadth\,of\,face\,2 \right)+............+ \\

 & \left( length\,of\,face\,6 \right)\times \left( breadth\,of\,face\,6 \right) \\

\end{align} \right]$

$\begin{align}

  & \Rightarrow TSA=\left[ \left( b\times h \right)+\left( b\times h \right)+\left( l\times b \right)+\left( l\times b \right)+\left( l\times h \right)+\left( l\times h \right) \right] \\

 & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\left[ 2bh+2lb+2lh \right] \\

\end{align}$

$\Rightarrow TSA=2\left( lb+bh+lh \right)$