The length of three unequal edges of a rectangular solid block are in G.P. The volume of the block is $216 cm^{3}$ and the total surface area is $252 cm^{2}$ . The length of the longest edge is ?
+128406 Let the edges be in order from shortest to longest a , ar and ar^2
So
The volume is
216 = a^3 r^3 ⇒ 216/r^3 = a^3 take the cube root 6/r = a
The suface area is
252 = 2 ( a* ar + a *ar^2 + ar * ar^2) simplify
252 = 2 ( a^2r + a^2r^2 + a^2r^3 )
126 = [ (6/r)^2 *r + (6/r)^2 *r^2 + (6/r)^2 * r^3 ]
126 = 36/r+ 36 + 36r multiply through by r
126r = 36 + 36r + 36r^2 rearrange as
36r^2 - 90r + 36 = 0 divide through by 18
2r^2 - 5r + 2 = 0 factor as
(2r - 1) ( r - 2) = 0
We have two solutions for r
r = 1/2 r = 2
The second is the one we need
r = 2 a = 6/r = 6/2 = 3
The edge lengths are a , ar , ar^2 = 2 , 2(3) , 2 (3)^2 = 3, 6 , 12
The longest edge is 12cm
