11.4. The base of a right prism is a quadrilateral inscribed in a circle with a radius of $25 \mathrm{~cm}$. The areas of the lateral faces are in the ratio 7:15:20:24, and the length of the diagonal of the largest lateral face is 52 cm. Calculate the surface area of the prism. (7 points)
# Solution

The sides of the base are $7 \mathrm{x}, 15 \mathrm{x}, 20 \mathrm{x}, 24 \mathrm{x}$. Since $(7 \mathrm{x})^{2}+(24 \mathrm{x})^{2}=(15 \mathrm{x})^{2}+(20 \mathrm{x})^{2}$, two angles of the base are right angles. $(15 \mathrm{x})^{2}+(20 \mathrm{x})^{2}=50^{2}, \mathrm{x}=2$. The sides of the base are $14,30,40,48$ cm, and the height $\mathrm{H}$ $=\sqrt{52^{2}-48^{2}}=20(\mathrm{~cm}) . \mathrm{Sn}=4512 \mathrm{~cm}^{2}$.

Answer: $4512 \mathrm{~cm}^{2}$

| criteria | points |
| :---: | :---: |
| 1. correct solution | 7 |


| 2. Proven that the diagonal of the base is the diameter | 4 |
| :--- | :---: |
| 3. height of the prism expressed as $\sqrt{52^{2}-(24 x)^{2}}$ | 2 |
| 4. sides of the base represented as $7 \mathrm{x}, 15 \mathrm{x}, 20 \mathrm{x}, 24 \mathrm{x}$ | 1 |
| 5. for each arithmetic error | -1 |