The curves $\vec{r}_{1}(t)=\left\langle 3 t, t^{4},-t^{5}\right\rangle$ and $\vec{r}_{2}(t)=\langle\sin (-3 t), \sin (-2 t), t-\pi\rangle$ intersect at the origin. Find the acute angle of intersection (in radians) on the domain $0 \leq \theta \leq \frac{\pi}{2}$, to at least two decimal places.

Step 1 ：Given the curves \(\vec{r}_{1}(t)=\left\langle 3 t, t^{4},-t^{5}\right\rangle\) and \(\vec{r}_{2}(t)=\langle\sin (-3 t), \sin (-2 t), t-\pi\rangle\), we need to find the acute angle of intersection at the origin.

Step 2 ：The angle of intersection between two curves at a point is the same as the angle between their tangent vectors at that point. So, we need to find the tangent vectors of the curves at the origin.

Step 3 ：The tangent vector of a curve \(\vec{r}(t)\) at a point \(t=t_0\) is given by the derivative \(\vec{r}'(t_0)\). So, we need to find \(\vec{r}_{1}'(0)\) and \(\vec{r}_{2}'(0)\).

Step 4 ：\(\vec{r}_{1}'(0)\) is \(\left\langle 3, 0, 0\right\rangle\) and \(\vec{r}_{2}'(0)\) is \(\left\langle -3, -2, 1\right\rangle\).

Step 5 ：The angle \(\theta\) between two vectors \(\vec{a}\) and \(\vec{b}\) is given by the formula \(\cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{||\vec{a}|| ||\vec{b}||}\), where \(\vec{a} \cdot \vec{b}\) is the dot product of \(\vec{a}\) and \(\vec{b}\), and \(||\vec{a}||\) and \(||\vec{b}||\) are the magnitudes of \(\vec{a}\) and \(\vec{b}\), respectively.

Step 6 ：The dot product of \(\vec{r}_{1}'(0)\) and \(\vec{r}_{2}'(0)\) is -9, the magnitude of \(\vec{r}_{1}'(0)\) is 3, and the magnitude of \(\vec{r}_{2}'(0)\) is \(\sqrt{14}\).

Step 7 ：Substituting these values into the formula, we get \(\cos(\theta) = \frac{-9}{3 \cdot \sqrt{14}} = -\frac{3 \sqrt{14}}{14}\).

Step 8 ：Taking the inverse cosine of this value, we get \(\theta = 2.50107034091037\) radians.

Step 9 ：However, we are asked to find the acute angle of intersection, which is on the domain \(0 \leq \theta \leq \frac{\pi}{2}\). Since \(\theta\) is greater than \(\frac{\pi}{2}\), we subtract it from \(\pi\) to get the acute angle, which is approximately 0.64052231267922 radians.

Step 10 ：Final Answer: The acute angle of intersection between the two curves at the origin is approximately \(\boxed{0.64}\) radians.