A unifrom square plate `S (side c)` and a unifrom rectangular plate `R(side b,a)` have identical areas and mass [Fig.]
Show that
(i) `I_(xR)//I_(xS) lt 1`, (ii) `I_(yR)//I_(yS) gt 1`, (iii) `I_(zR)//I_(zS) gt 1`.

Explanation

As area of square plate `S = area` of rectangular plate `R`
`c^(2) = a xx b`
Now, by definition
(i) `(I_(xR))/(I_(xS)) = (b^(2))/(c^(2))`
As `b lt c, (I_(xR))/(I_(xS)) lt 1`
(ii) `(I_(yR))/(I_(yS)) = (a^(2))/(c^(2))`
As `a gt c, (I_(yR))/(I_(yS)) gt 1`.
(iii) `(I_(zR) - I_(zS)) prop (a^(2) + b^(2) - 2c^(2))`
`prop (a^(2) + b^(2) - 2 ab)`
`prop (a - b)^(2)`, which is +
`:. (I_(xR) I_(xS)) gt 0 or (I_(xR))/(I_(xS)) gt 1`