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# solid geo

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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

Dec 27, 2020