+0  
 
0
51
1
avatar

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 ?

 Feb 1, 2021
 #1
avatar+116126 
+1

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 

 

 

cool cool cool

 Feb 1, 2021

52 Online Users

avatar
avatar