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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