10,11 In a rectangular parallelepiped $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime} A B=B C=a, A A^{\prime}=b$. It was orthogonally projected onto a certain plane containing the edge $C D$. Find the maximum value of the projection area. #
- Let the projection plane $\alpha$ be located as shown in the figure. Then the projection of the parallelepiped coincides with the projection of its diagonal section $A B C^{\prime} D^{\prime}$. Since the area of the orthogonal projection of a polygon does not exceed the area of the polygon, the area of the projection will be the largest if $\alpha$ is parallel to this section. In this case, $S_{\text {proj }}=S_{A B C^{\prime} D^{\prime}}=A B \cdot A D^{\prime}=a \sqrt{a^{2}+b^{2}}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_43a9f723b796fae20c45g-19.jpg?height=632&width=554&top_left_y=233&top_left_x=769)

If the plane $\alpha$ contains internal points of the parallelepiped, then its projection coincides with the projection of the diagonal section $A^{\prime} B^{\prime} C D$. Since the areas of the diagonal sections are equal, the answer will not change.

## Answer

$$
a \sqrt{a^{2}+b^{2}}
$$