A solid cylinder has total surface area of $462 cm^2$. Its curved surface area is one-third of its total surface are. Find the volume of the cylinder (Take $\pi=\cfrac{22}{7}$)

Using the given total surface area of 462 square cm, the curved surface area is one-third of this, which equals 154 square cm. Subtracting the curved surface area from the total surface area leaves 308 square cm for the combined area of the two circular bases, meaning one base has an area of 154 square cm. Using the area formula pi times radius squared, we get twenty-two sevenths times radius squared equals 154, so the radius squared is 49 and the radius is 7 cm. Since the curved surface area equals 2 times pi times radius times height, we have 154 equals 2 times twenty-two sevenths times 7 times height, making the height 3.5 cm. The volume is pi times radius squared times height, which is twenty-two sevenths times 49 times 3.5, totaling 539 cubic cm; therefore, none of the listed numerical options are correct.

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