1. **State the problem:** We have a cylinder net with two circular bases each of radius 5 mm and a rectangular side with height 4 mm and unknown length $x$. We need to find $x$ and then calculate the total surface area of the cylinder in terms of $\pi$. 2. **Formula for the circumference of a circle:** The length $x$ corresponds to the circumference of the circular base because the rectangle wraps around the circle. $$x = 2\pi r$$ where $r$ is the radius. 3. **Calculate $x$:** Given $r = 5$ mm, $$x = 2\pi \times 5 = 10\pi \text{ mm}$$ 4. **Formula for total surface area of a cylinder:** The total surface area $A$ is the sum of the areas of the two circular bases and the rectangular side (lateral surface area): $$A = 2\pi r^2 + 2\pi r h$$ where $h$ is the height of the cylinder. 5. **Calculate the total surface area:** Given $r = 5$ mm and $h = 4$ mm, $$A = 2\pi (5)^2 + 2\pi (5)(4) = 2\pi (25) + 2\pi (20) = 50\pi + 40\pi = 90\pi \text{ mm}^2$$ **Final answers:** - Length $x = 10\pi$ mm - Total surface area $= 90\pi$ mm$^2$