Let $K$ be a cube of side $1$ in $\mathbb{R}^{3}$, with its centre at the origin, and its sides parallel to the co-ordinate axes. For $t \geq 0$, let $K_{t}$ be the set of all points in $\mathbb{R}^{3}$ whose Euclidean distance to $K$ is less than or equal to $t$. Let $V_{t}$ be the volume of $K_{t}$. Then, which of the following is $\text{TRUE}$ for all $t \geq 0$ ?
Note: If $d(x, y)$ denotes the Euclidean distance between two points $x$ and $y$ in $\mathbb{R}^{3}$, then the distance of a point $p \in \mathbb{R}^{3}$ to $K$ is defined as the minimum of the quantities $d(p, y)$ when $y$ ranges over $K$.
- $V_{t} \leq 1$
- $V \leq 1+\frac{4}{3} \pi t^{3}$
- $V \leq 1+\frac{4}{3} \pi t$
- $V \leq \frac{4}{3} \pi\left(\frac{\sqrt{3}}{2}+t\right)^{3}$
- $V \geq(1+2 t)^{3}$