At time $t$, a particle moving in the $xy$-plane is at position $(x(t), y(t))$, where $x(t)$ and $y(t)$ are not explicitly given. For $t \geq 0$, $\frac{dx}{dt} = 4t + 1$ and $\frac{dy}{dt} = \sin\left(t^2\right)$. At time $t = 0$, $x(0) = 0$ and $y(0) = -4$.
(a) Find the speed of the particle at time $t = 3$, and find the acceleration vector of the particle at time $t = 3$.
(b) Find the slope of the line tangent to the path of the particle at time $t = 3$.
(c) Find the position of the particle at time $t = 3$.
(d) Find the total distance traveled by the particle over the time interval $0 \leq t \leq 3$.