Question

The length and breadth and height of a rectangular solid are in the ratio of 5:4:2. If the total surface area is $1216c{{m}^{2}}$, find the length of the solid in cm.

Answer 1

Hint: If the length, breadth and height of any cuboid is given as l, b and h, then the total surface area of the cuboid is given as ‘S = 2(lb+bh+hl)’ where ‘S’ is denoting the total surface area. Correspondingly find the length of the solid.

Complete step-by-step answer:
Let us use variable ‘x’ for writing the length, breadth and height of the rectangular solid.
As the ratio of sides is 5:4:2, then let the length, breadth and height be $5x,4x,2x$, respectively.

Now, we know the total surface area of any cuboid (rectangular solid) is given as
S = 2(lb+bh+hl)…………….(i)

Here ‘l’ is the length of cuboid, ‘b’ is breadth of cuboid, ‘h’ is height of cuboid and ‘S’ is representing the total surface area.

So, we have ‘l = 5x’, ‘b = 4x’ and ‘h = 2x’ from the given problem.

Hence, total surface area can be given using the relation (i) as,
$\begin{align}
  & S=2\left( 5x\times 4x+4x\times 2x+2x\times 5x \right) \\
 & S=2\left( 20{{x}^{2}}+2{{x}^{2}}+10{{x}^{2}} \right) \\
 & S=2\left( 38{{x}^{2}} \right) \\
 & S=76{{x}^{2}}c{{m}^{2}}....................\left( ii \right) \\
\end{align}$

Now it is given in the question that the rectangular solid has a total surface area of $1216c{{m}^{2}}$ . So, we can equate equation (ii) with this value, we get
$\begin{align}
  & S=76{{x}^{2}}=1216 \\
 & \Rightarrow {{x}^{2}}=\dfrac{1216}{76}=16 \\
\end{align}$

Taking square root on both sides, we get
$x=\pm 4$

We cannot take ‘x = -4’ as the sides of any other rectangular solid can never be negative.

Hence, the value of ‘x’ would be 4.

So, length, breadth, height of the given rectangular solid can be given as
Length $=5x=5\times 4=20cm$
Breadth $=4x=4\times 4=16cm$
Height $=2x=2\times 4=8cm$

Hence, the length of the solid is 20cm.

Note: One can prove the formula of total surface area by adding the area of all the faces of cuboid. Don’t use formula $2\left( l+b \right)\times h$ as it is containing the area of upper and lower face. So, be careful with the terms and concepts both.
One can use any variable for writing the sides of rectangular solid. Writing the sides with the help of a variable is the key point of the question.