Answer
$$
\begin{aligned}
& \mathbf{v}_A=\{-5.20 \mathbf{i}-12 \mathbf{j}+20.8 \mathbf{k}\} \mathrm{ft} / \mathrm{s} \\
& \mathbf{a}_A=\{-3.33 \mathbf{i}-21.3 \mathbf{j}+6.66 \mathbf{k}\} \mathrm{ft} / \mathrm{s}^2
\end{aligned}
$$
Work Step by Step
$$
\begin{aligned}
\mathbf{r}_{A I O}= & 40 \cos 30^{\circ} \mathbf{j}+40 \sin 30 \mathbf{k} \\
\mathbf{r}_{A O O}= & \{34.641 \mathbf{j}+20 \mathbf{k}\} \mathrm{ft} \\
\Omega= & \omega_1 \mathbf{k}+\omega_2 \mathbf{i}=\{0.6 \mathbf{i}+0.15 \mathbf{k}\} \mathrm{rad} / \mathrm{s} \\
\dot{\omega}= & \dot{\omega}_1 \mathbf{k}+\dot{\omega}_2 \mathbf{i}+\omega_1 \mathbf{k} \times \omega_2 \mathbf{i} \\
\dot{\Omega}= & 0.2 \mathbf{k}+0.4 \mathbf{i}+0.15 \mathbf{k} \times 0.6 \mathbf{i}=\{0.4 \mathbf{i}+0.09 \mathbf{j}+0.2 \mathbf{k}\} \mathrm{rad} / \mathrm{s}^2 \\
\mathbf{v}_A= & \Omega \times \mathbf{r}_{A O}=(0.6 \mathbf{i}+0.15 \mathbf{k}) \times(34.641 \mathbf{j}+20 \mathbf{k}) \\
\mathbf{v}_A= & \{-5.20 \mathbf{i}-12 \mathbf{j}+20.8 \mathbf{k}\} \mathrm{ft} / \mathrm{s} \\
\mathbf{a}_A= & \Omega \times\left(\Omega \times \mathbf{r}_{A O}\right)+\dot{\omega} \times \mathbf{r} \mathbf{r}_{A O} \\
\mathbf{a}_A= & (0.6 \mathbf{i}+0.15 \mathbf{k}) \times[(0.6 \mathbf{i}+0.15 \mathbf{k}) \times(34.641 \mathbf{j}+20 \mathbf{k})] \\
& +(0.4 \mathbf{i}+0.09 \mathbf{j}+0.2 \mathbf{k}) \times(34.641 \mathbf{j}+20 \mathbf{k}) \\
\mathbf{a}_A= & \{-3.33 \mathbf{i}-21.3 \mathbf{j}+6.66 \mathbf{k}\} \mathrm{ft} / \mathrm{s}^2
\end{aligned}
$$
