The radius of a right cylinder having a height of $15\text{ cm}$ and a surface area of $U{\text{ cm}}^2$ is given as $r\left(U\right)=\frac{1}{5}(\sqrt{225+\frac{5U}{\pi }}-15)$. Calculate $\underset{U\to 0^{+}}{\lim }r\left(U\right)$ and provide an interpretation.

• A. Limit: $\frac{1}{2}\sqrt{105}$Interpretation: As $U$ approaches $0$ from the positive side, the radius of the cylinder approaches $\frac{1}{2}\sqrt{105}$. This indicates that for a very small surface area, the radius of the cylinder will be approximately $\frac{1}{2}\sqrt{105}$.
• B. Limit: $\frac{1}{5}\sqrt{21}$Interpretation: As $U$ approaches $0$ from the positive side, the radius of the cylinder approaches $\frac{1}{5}\sqrt{21}$. This indicates that for a very small surface area, the radius of the cylinder will be approximately $\frac{1}{5}\sqrt{21}$.
• C. Limit: $0$Interpretation: As $U$ approaches $0$ from the positive side, the radius of the cylinder approaches $0$. This indicates that for a very small surface area, the radius of the cylinder will be approximately $0$.
• D. Limit: $\frac{1}{2}\sqrt{210}$Interpretation: As $U$ approaches $0$ from the positive side, the radius of the cylinder approaches $\frac{1}{2}\sqrt{210}$. This indicates that for a very small surface area, the radius of the cylinder will be approximately $\frac{1}{2}\sqrt{210}$.

Answer: (C)