Question : The breadth ‘b’ of a room is twice its height and half of its length. Find the length of the longest diagonal of the room.
Option 1: $\frac{\sqrt{20} \mathrm{~b}}{2}$
Option 2: $\frac{\mathrm{b}}{2}$
Option 3: $\frac{\sqrt{21} \mathrm{~b}}{2}$
Option 4: $\frac{\sqrt{19} \mathrm{~b}}{2}$
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Correct Answer: $\frac{\sqrt{21} \mathrm{~b}}{2}$
Solution :
According to the question,
b = 2 × height
⇒ Height = $\frac{1}{2}$b
Also, b = $\frac12\times$length
⇒ Length = 2b
Length of the diagonal
$=\sqrt{l^2+b^2+h^2}$
$=\sqrt{(2b)^2+b^2+(\frac12b)^2}$
$=\sqrt{4b^2+b^2+\frac {b^2}{4}}$
$=\sqrt{\frac{16b^2+4b^2+b^2}{4}}$
$=\sqrt{\frac{21b^2}{4}}$
$=\frac{\sqrt{21}b}{2}$
Hence, the correct answer is $\frac{\sqrt{21}b}{2}$.
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