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.
+128474 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
