The three different face diagonals of a cuboid (rectangular parallelopipd ) have lengths 39,40 and 41 units .Find the length of the main diagonal of the cuboid which joins a pair of opposite corners.

Explanation

Let l,b,h be the length ,breadth and height of a cuboid.
`(sqrt(l^(2)+b^(2)))=39rarr l`
`(sqrt(b^(2)+h^(2)))=40 rarr b^(2)+h^(2)=(40)^(2)`=1600..(2)
`(sqrt(h^(2)+l^(2)))=4lrarrh^(2)+l^(2)=(41)^(2)=168…(3)`
Adding (1),(2)and(3), we get
`2(l^(2)+b^(2)+h^(2))=1521+1600+1681=4802`
`l^(2)+b^(2)+h^(2)=2401=(49^(2))`
Length of main diagonal EB=`(sqrt(l^(2))+b^(2)+h^(2)))=(sqrt(49^(2)))=49 units`