Question

Inner part of a cupboard is in the cuboidical shape with its length, breadth and height in the ratio $1:\sqrt{2}:1$. What is the angle made by the longest stick which can be inserted cupboard with its base inside.

Answer 1

Length: Breadth:Height $=1:\sqrt{2}:1$

$\Rightarrow$ length $=x=BC$

Breadth$=\sqrt{2}x=AB$

Height $=x=AE$

Longest stick in the cupboard is shown as $EC$

In $\Delta ABC$,

Applying Pythagoras theorem,

$A{C}^2=A{B}^2+B{C}^2$
$A{C}^2=(\sqrt{2}x{)}^2+x^2$
$2x^2+x^2=3x^2$
$AC=\sqrt{3}x$

In $\Delta ACE$,

$\tan \theta =\frac{AE}{AC}=\frac{x}{\sqrt{3}x}=\frac{1}{\sqrt{3}}$
$\Rightarrow \theta =30°$

Ans ${30}^o$