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Shreya got a rectangular parallelepiped shaped box and spherical ball inside it as return gift. Sides of the box are $x, 2x,$ and $\frac{\text{x}}{3},$ while radius of the ball is $r.$

Based on the above information, answer the following questions.

1. If S represents the sum of volume of parallelepiped and sphere, then Scan be written as.

1. $\frac{4\text{x}^3}{3}+\frac{2}{2}\pi\text{r}^2$
2. $\frac{2\text{x}^2}{3}+\frac{4}{3}\pi\text{r}^2$
3. $\frac{2\text{x}^3}{3}+\frac{4}{3}\pi\text{r}^3$
4. $\frac{2}{3}\text{x}+\frac{4}{3}\pi\text{r}$

2. If sum of the surface areas of box and ball are given to be constant $k^2$ then $x$ is equal to.

1. $\sqrt{\frac{\text{k}^2-4\pi\text{r}^2}{6}}$
2. $\sqrt{\frac{\text{k}^2-4\pi\text{r}}{6}}$
3. $\sqrt{\frac{\text{k}^2-4\pi}{6}}$
4. $\text{None of these}$

3. The radius of the ball, when Sis minimum, is.

1. $\sqrt{\frac{\text{k}^2}{54+\pi}}$
2. $\sqrt{\frac{\text{k}^2}{54+4}}$
3. $\sqrt{\frac{\text{k}^2}{64+3\pi}}$
4. $\sqrt{\frac{\text{k}^2}{4\pi+3}}$

4. Relation between length of the box and radius of the ball can be represented as.

1. $\text{x} = \frac{2}{\text{r}}$
2. $\text{x}=\frac{\text{r}}{2}$
3. $\text{x}=\frac{2}{\text{r}}$
4. $\text{x}=3\text{r}$

5. Minimum value of $S$ is.

1. $\frac{\text{k}^2}{2(3\pi+54)^\frac{2}{3}}$
2. $\frac{\text{k}}{2(3\pi+54)^\frac{3}{2}}$
3. $\frac{\text{k}^3}{2(4\pi+54)^\frac{1}{2}}$
4. $\text{None of these}$