5. The base of a right prism is a rectangle with sides $5 \mathrm{~cm}$ and $8 \mathrm{~cm}$, and the surface area of the lateral face of this prism is equal to $408.2 \mathrm{~cm}^{2}$. Determine the surface area of the lateral face of a cylinder whose base radius is $4 \mathrm{~cm}$, if the volume of that cylinder is $64 \%$ of the volume of the given prism. (Take $\pi \approx 3.14$.)
Solution. Let $M_{p}$ be the area of the lateral surface, $O$ be the perimeter of the base of the prism, and $H$ be its height. We have $M_{p}=O \cdot H$, so $408.2=26 H$, i.e., $H=\frac{408.2}{26} \text{ cm}$. Therefore, the volume of the prism is $V_{p}=B \cdot H=8 \cdot 5 \cdot \frac{408.2}{26} \text{ cm}^{3}$, and the volume of the cylinder is $V_{C}=\frac{64}{100} V_{p}=\frac{64 \cdot 8 \cdot 5 \cdot 408.2}{100 \cdot 26} \text{ cm}^{3}$. However, $V_{C}=\pi r^{2} H_{C}=16 \pi H_{C}$, so $16 \pi H_{C}=\frac{64 \cdot 8 \cdot 5 \cdot 408.2}{100 \cdot 26}$, i.e., $H_{C}=\frac{64 \cdot 8 \cdot 5 \cdot 408.2}{100 \cdot 26 \cdot 16 \pi} \text{ cm}$. The area of the lateral surface of the cylinder is:

$$
M_{C}=2 \pi r H_{C}=2 \pi \cdot 4 \cdot \frac{64 \cdot 8 \cdot 5 \cdot 408.2}{100 \cdot 26 \cdot 16 \pi}=200.96 \text{ cm}^{2}
$$