Question

The length, breadth and height of the rectangular box are in the ratio of 4:3:2. The total surface area of the box is \[2548c{m^2}.\] Find its Length breadth and height.

Answer 1

Hint: As the dimensions are in ratio assume it by multiplying with a common variable and also remember that the surface area of a cuboid is \[2(lb + bh + lh)\] where l stands for length b stands for breadth and h stands for height.
Complete step by step answer:
Let us assume that x is the variable and the common factor between length breadth and height So Now the dimensions of the Cuboid becomes
\[\begin{array}{l}
Length = 4x\\
Breadth = 3x\\
Height = 2x
\end{array}\]
Now using all of this let us try to formulate an equation in x.
We know,
\[\begin{array}{l}
\therefore Area = 2(lb + bh + lh)\\
 \Rightarrow 2548 = 2(lb + bh + lh)\\
 \Rightarrow \dfrac{{2548}}{2} = lb + bh + lh\\
 \Rightarrow (4x \times 3x) + (3x \times 2x) + (4x \times 2x) = 1274\\
 \Rightarrow 12{x^2} + 6{x^2} + 8{x^2} = 1274\\
 \Rightarrow 26{x^2} = 1274\\
 \Rightarrow {x^2} = \dfrac{{1274}}{{26}}\\
 \Rightarrow {x^2} = 49\\
 \Rightarrow x = \sqrt {49} \\
 \Rightarrow x = \pm 7
\end{array}\]
 So we are getting our common factor as 7.
Now Putting this in the values of length breadth and height we get it as
\[\begin{array}{*{20}{l}}
{Length = 4x = 4 \times 7 = 28}\\
{Breadth = 3x = 3 \times 7 = 21}\\
{Height = 2x = 2 \times 7 = 14}
\end{array}\]
Now it is clear that the dimensions of the cuboid are 28cm, 21cm, 14cm.

Note: It must be known that a rectangle in 2-dimension is cuboid in 3-dimension, Similarly a circle in 2D is a Sphere in 3D also a Square in 2D is a Cube in 3D. Also dimensions cannot be in negative, that’s why -7 was not taken under account, only +7 was used.