If the radius of a cylinder increases by $y$ cm, its volume becomes $V$ $\displaystyle cm^{3}$ . If its height increased by $4y$ cm, its volume would become $V$ $\displaystyle cm^{3}$. If its radius as well as its height equals $6$ cm, then find $y$.

The initial radius and height are both 6 cm, making the original volume V equal to 216(pi). The first condition states V = (pi)(6 + y)^2 times 6, meaning 216 = 6 times (6 + y)^2, which simplifies to (6 + y)^2 = 36. Solving this equation gives a y value of 12, and this is confirmed by the second condition where 216(pi) equals (pi)(6^2) times (6 + 4y), leading to the exact same result. The value of y is 12.

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