The door of the rabbit hutch is made of a wooden frame and wire mesh with square holes. The frame strips are $5 \mathrm{~cm}$ wide. Some nodes of the mesh align with the inner edges of the frame as shown in the picture. The inner (mesh) part of the door has an area of $432 \mathrm{~cm}^{2}$.

Determine the external dimensions (i.e., width and height) of the entire door.

( $S$. Bednářová)

![](https://cdn.mathpix.com/cropped/2024_04_17_a7ae5c887c970a6f6dc9g-2.jpg?height=568&width=462&top_left_y=504&top_left_x=800)
The mesh part of the door consists of identical squares and their halves. The inner edges of the frame are formed by the diagonals of these squares, with three on the shorter side and four on the longer side. If \( a \) is the length of the diagonal of a square, the mesh part has dimensions \( 3a \times 4a \).

![](https://cdn.mathpix.com/cropped/2024_04_17_a7ae5c887c970a6f6dc9g-2.jpg?height=574&width=479&top_left_y=1369&top_left_x=797)

This corresponds to an area of \( 432 \, \text{cm}^2 \), thus

\[
12a^2 = 432, \quad a^2 = 36, \quad a = 6.
\]

The inner dimensions of the door are \( 18 \, \text{cm} \times 24 \, \text{cm} \), and the outer dimensions are \( 28 \, \text{cm} \times 34 \, \text{cm} \) (5 cm added to each side).

Note. The mesh part of the door consists of 17 identical (whole) squares and 14 halves of these squares. In total, there are 24 squares, which have an area of \( 432 \, \text{cm}^2 \). Therefore, each square has an area of \( 18 \, \text{cm}^2 \). The area of a square with diagonal length \( a \) is \( \frac{1}{2}a^2 \). The diagonal of the mesh square can thus be determined by the following adjustments:

\[
\frac{1}{2}a^2 = 18, \quad a^2 = 36, \quad a = 6.
\]

Evaluation. 2 points for auxiliary decompositions and considerations; 2 points for auxiliary calculations; 2 points for the conclusion and quality of the commentary.