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,
$$\begin{array}{l}\therefore Area=2(lb+bh+lh)\\ \Rightarrow 2548=2(lb+bh+lh)\\ \Rightarrow {\displaystyle \frac{2548}{2}}=lb+bh+lh\\ \Rightarrow (4x\times 3x)+(3x\times 2x)+(4x\times 2x)=1274\\ \Rightarrow 12x^2+6x^2+8x^2=1274\\ \Rightarrow 26x^2=1274\\ \Rightarrow x^2={\displaystyle \frac{1274}{26}}\\ \Rightarrow x^2=49\\ \Rightarrow x=\sqrt{49}\\ \Rightarrow x=\pm 7\end{array}$$
 So we are getting our common factor as 7.
Now Putting this in the values of length breadth and height we get it as
$$\begin{array}{l}Length=4x=4\times 7=28\\ Breadth=3x=3\times 7=21\\ Height=2x=2\times 7=14\end{array}$$
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.