Increasing the radius of a cylinder by 6 units increases the volume by y cubic units. Increasing the altitude of the cylinders by 6 units also increases the volume by y cubic units. If the original altitude is 2, then the original radius is

Let the original radius be r and the original height be 2; the volume increase from adding 6 to the radius is pi * (r + 6)^2 * 2 - pi * r^2 * 2, which equals 24 * pi * r + 72 * pi. The volume increase from adding 6 to the height is pi * r^2 * 8 - pi * r^2 * 2, which equals 6 * pi * r^2. Setting the increases equal gives 6 * r^2 = 24 * r + 72, and dividing by 6 results in r^2 - 4r - 12 = 0; factoring this gives (r - 6)(r + 2) = 0, so the original radius is 6.

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