Question #41695

Let $\hat{n} = \frac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k})$. A plane $\Pi$ in $\mathbb{R}^3$ is defined by the equation $\vec{r} \cdot \hat{n} = 2\sqrt{3}$. A feasible region $S$ on $\Pi$ is bounded by the non-negativity conditions $\vec{r} \cdot \hat{i} \ge 0$, $\vec{r} \cdot \hat{j} \ge 0$, $\vec{r} \cdot \hat{k} \ge 0$, and the following projection constraints: $\vec{r} \cdot (\hat{i} + 2\hat{j}) \le 8$ $\vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) \le 14$ Determine the maximum value of the objective function $Z = \vec{r} \cdot (10\hat{i} + 24\hat{j} + 8\hat{k})$ for any position vector $\vec{r} \in S$.