The length, breadth and height of a cuboid room be a unit, b unit and c unit respectively and a + b + c = 25, ab + bc + bc = 240.5 then find the length of the longest rod to be kept inside the room.

Solution :

The length of the longest rod should be equal to the length of the diagonal of the cuboidal room.

**∴** The required length of the longest rod is given by

L = √ ( a^{2} + b^{2} + c^{2} )

= √ { ( a^{2} + b^{2} + c^{2} ) – ( ab + bc + ca) }

[ ∵ ( a^{2} + b^{2} + c^{2} ) = a^{2} + b^{2} + c^{2} + ab + bc + ca

⇒ ( a^{2} + b^{2} + c^{2} ) = ( a + b + c )^{2} – ab – bc – ca

⇒ ( a^{2} + b^{2} + c^{2} ) = ( a + b + c )^{2} – ( ab – bc – ca ) ]

= √ { ( 25)^{2} – 240.5 }

= √ { 625 – 240.5 }

= √ ( 384.5 )

= √ ( 3845 / 10)

= 62 / √ ( 10 ) cm

= 62 / 3.16 [ ∵ √( 10 ) **≃** 3.16 ]

= 19.62 cm ( Approximately ).

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