The length, breadth and height of a cuboid of a 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. ( WBBSE – 2019 )

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 ).