Final answer:
The problem deals with the calculation of the height of a cylinder with a specific volume, given that its height is six times its radius. The radius is calculated to be approximately 4.5 dcm, thus, the height of the cylinder is approximately 27 dcm.
Step-by-step explanation:
The subject of this problem is a right circular cylinder whose height is initially twice its radius. The volume V of such a cylinder is given by the formula V = πr²h, where r represents the radius and h represents the height of the cylinder.
If the height is twice the radius (h = 2r), the volume is V = πr²(2r) = 2πr³. If the height is six times the radius (h = 6r), the volume is V' = πr²(6r) = 6πr³.
The increase in the volume then, when the height is made six times the radius instead of twice the radius, is ΔV = V' - V = 6πr³ - 2πr³ = 4πr³.
This increase in volume, ΔV, is given as 539 cubic dcm. So we set 4πr³ = 539 and solve for r. That calculation leads to a radius of approximately 4.5 dcm. This radius is necessary to achieve the increased volume. Hence, the height of the cylinder, which is six times the radius, is approximately 27 dcm.
Learn more about Volume of a Cylinder