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\(\right\))=\(\frac\)15\(\left\)(\(\sqrt{225+\frac{5U}{\pi}\)}-15\(\right\))$. Calculate ${\(\displaystyle\]\lim\)_{U\(\to\)0^{+}}}r\(\left\)(U\(\right\))$ and provide an interpretation.

$\(\frac\)12\(\sqrt{105}\)$.

Interpretation: As U approaches 0 from the positive side, the radius of the cylinder approaches $\(\frac\)12\(\sqrt{105}\)$. This indicates that for a very small surface area, the radius of the cylinder will be approximately $\(\frac\)12\(\sqrt{105}\)$.

$\(\frac\)15\(\sqrt{21}\)$.

Interpretation: As U approaches 0 from the positive side, the radius of the cylinder approaches $\(\frac\)15\(\sqrt{21}\)$. This indicates that for a very small surface area, the radius of the cylinder will be approximately $\(\frac\)15\(\sqrt{21}\)$.

$\(\frac\)15\(\sqrt{210}\)$.

Interpretation: As U approaches 0 from the positive side, the radius of the cylinder approaches $\(\frac{1}{5}\) \(\sqrt{210}\)$. This indicates that for a very small surface area, the radius of the cylinder will be approximately $\(\frac{1}{5}\) \(\sqrt{210}\)$.

$\(\frac\)12\(\sqrt{210}\)$.

Interpretation: As U approaches 0 from the positive side, the radius of the cylinder approaches $\(\frac\)12\(\sqrt{210}\)$. This indicates that for a very small surface area, the radius of the cylinder will be approximately $\(\frac\)12\(\sqrt{210}\)$.