On increasing the radius of a cylinder by 9 units, the volume increases by (p) cubic units. On increasing the altitude of the cylinder by 9 units, the volume also increases by (p) cubic units. If the original altitude is 3 units, what is the original radius?

The volume of a cylinder is V = πr²h. When only the radius increases by 9: V₁ = π(r+9)²(3) = 3π(r²+18r+81), so p = 54πr + 243π. When only the height increases by 9: V₂ = πr²(12) = 12πr², so p = 9πr². Setting equal: 54πr + 243π = 9πr². Dividing by 9π: 6r + 27 = r², so r² - 6r - 27 = 0. Factoring: (r-9)(r+3) = 0. Since radius must be positive, r = 9 units.