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Two congruent cylinders each have radius 8 inches and height 8 inches. The radius of one cylinder and the height of the other are both increased by the same number of inches. The resulting volumes are equal. How many inches is the increase? Express your answer as a common fraction.

 Dec 27, 2020
 #1
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This is impossible

 

To see  why, let  the increase in inches  =  a    where a >  0

 

Volume of the  cylinder  with  the radius increase will be

 

pi ( r + a)^2 * h  =   pi  (r^2  + 2ar  + a^2) * h =  pi (r^2h + 2arh + a^2h)

 

Volume of cylinder  with height increase

 

pi (r)^2 (h + a) =   pi  ( r^2 h + ar^2)

 

So  equal   volumes means that

 

pi ( r^2h + 2arh  + a^2h)  = pi ( r^2 h + ar^2)      divide out pi

 

r^2 h + 2arh + a^2h  = r^2h + ar^2        subtract  r^2 h  from both sides

 

2arh + a^2h  =  ar^2        since a > 0, divide out  a

 

2rh  + ah  =  r^2        rearrange as

 

ah  =  r^2  - 2rh     factor

 

ah = r ( r - 2h)

 

But   r =  h   = 8     

 

So

 

8a  =  8 ( 8  - 16)     divide out 8

 

a =  -8

 

But we  have made the assumption that  a > 0....so.....we have a  contradiction

 

 

cool cool cool

 Dec 27, 2020

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