Question

If the lengths of the three unequal edges of a rectangular solid block are in G.P., volume of the block is $216$ ${\text{cm}}^3$ and its total surface area is $252{\text{cm}}^2,$ then the length of the longest edge (in cm) is equal to

Answer 1

Let $x,y,z$ be the three unequal edges $(x>y>z)$
Given, $x,y,z$ are in G.P.
$\Rightarrow y^2=xz\dots \left(1\right)$
Volume, $xyz=216\dots \left(2\right)$
From $\left(1\right)$ and $\left(2\right)$
$\Rightarrow y^3=216\Rightarrow y=6$
Total surface area, $2(xy+yz+zx)=252$
$\Rightarrow y(x+z)+xz=126$
$\Rightarrow 6(x+z)+36=126(\because xz=y^2=36)$
$\Rightarrow x+z=15$ and $xz=36$
$\Rightarrow x=12,z=3$
$\therefore$ Length of the longest edge is $12$ cm