Question

The length, breadth and height of a cuboid are in the ratio 5 : 4 : 2 and the total surface area is 1216 \[c{{m}^{2}}\], then the volume of the cuboid is:
A.2460 $c{{m}^{3}}$
B.2560 $c{{m}^{3}}$
C.2660 $c{{m}^{3}}$
D.2700 $c{{m}^{3}}$

Answer 1

Hint: In the above question assume the length, breadth and height as 5x, 4x and 2x of the given cuboid and find the value of x by equating the given total surface area. Now use the formula of the volume of the cuboid and find its volume. The formulae of the total surface area and the volume of the cuboid are as follows:

Complete step-by-step answer:
Let us consider the length, breadth and height of a cuboid to be l, b and h. Then,
\[\begin{align}
  & \text{Total surface area = 2}\left( l.b+b.h+l.h \right)\text{ sq}\text{. units} \\
 & \text{Volume = }l\times b\times h\text{ cubic units} \\
\end{align}\]

Now, we have been given that the length, breadth and height of a cuboid are in the ratio 5 : 4 : 2 and total surface area is 1216\[c{{m}^{2}}\].
Let us assume the length, breadth and height of a cuboid are 5x, 4x and 2x respectively. So, we have the figure as

Then, we can write the total surface area = \[2\left( \left( 5x\times 4x \right)+\left( 4x\times 2x \right)+\left( 5x\times 2x \right) \right)\]
\[\begin{align}
  & \Rightarrow 1216=2\left( 20{{x}^{2}}+8{{x}^{2}}+10{{x}^{2}} \right) \\
 & \Rightarrow 1216=2\left( 38{{x}^{2}} \right) \\
 & \Rightarrow \dfrac{1216}{2\times 38}={{x}^{2}} \\
 & \Rightarrow 16={{x}^{2}} \\
 & \Rightarrow x=4 \\
\end{align}\]
So, the length, breadth and height of the cuboid after substituting the value of x, are as follows:
Length = 20 cm
Breadth = 16 cm
Height = 8 cm
Now, the volume of the cuboid = \[20\times 16\times 8c{{m}^{3}}=2560c{{m}^{3}}\]
Therefore, the correct option of the above question is option B.

Note: Just be careful while doing calculation as there is a chance that you might make a mistake and you will get the incorrect answer. You should read the question carefully, by mistake if you read it as volume is given as 1216, then you will formulate the wrong equation and get a different value of x.