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# plz help

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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+0 Answers

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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   Feb 1, 2021