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 = √ ( a2 + b2 + c2 )

= √ { ( a2 + b2 + c2 ) – ( ab + bc + ca) }

[ ∵ ( a2 + b2 + c2 ) = a2 + b2 + c2 + ab + bc + ca

⇒ ( a2 + b2 + c2 ) = ( a + b + c )2 – ab – bc – ca

⇒ ( a2 + b2 + c2 ) = ( 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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