Solution:

Given the position vector:

\[
\mathbf{r} = (3t)\hat{i} - (t^2)\hat{j} + 4\hat{k}
\]

To find the velocity, differentiate the position vector with respect to time \(t\):

\[
\mathbf{v} = \frac{d\mathbf{r}}{dt} = \left( \frac{d}{dt}(3t) \hat{i} + \frac{d}{dt}(-t^2) \hat{j} + \frac{d}{dt}(4) \hat{k} \right)
\]

Calculating the derivatives:

\[
\mathbf{v} = (3)\hat{i} - (2t)\hat{j} + (0)\hat{k} = 3\hat{i} - 2t\hat{j}
\]

Now, substitute \(t = 5\) s:

\[
\mathbf{v}(5) = 3\hat{i} - 2(5)\hat{j} = 3\hat{i} - 10\hat{j}
\]

Calculate the magnitude of the velocity:

\[
|\mathbf{v}| = \sqrt{(3)^2 + (-10)^2} = \sqrt{9 + 100} = \sqrt{109}
\]

Thus, the magnitude of velocity after 5 seconds is: 10.44 m/s