1. **Problem Statement:** We have a concrete beam in the shape of a rectangular prism resting on two closed cylinders (pillars). We need to find: - (2.1.1) The surface area of one closed cylinder. - (2.1.2) The total volume of concrete needed to make the beam and the two pillars. 2. **Given Data:** - Beam dimensions: length $L=12$ cm, width $B=8$ cm, height $h=6$ cm. - Pillars: each is a closed cylinder with height $H=8$ cm and diameter $3$ cm, so radius $r=\frac{3}{2}=1.5$ cm. --- 3. **Step 2.1.1: Surface Area of One Closed Cylinder** - Formula for surface area of a closed cylinder: $$S = 2\pi r^2 + 2\pi r H$$ where $2\pi r^2$ is the area of the two circular bases and $2\pi r H$ is the lateral surface area. - Substitute values: $$S = 2\pi (1.5)^2 + 2\pi (1.5)(8)$$ $$= 2\pi (2.25) + 2\pi (12)$$ $$= 4.5\pi + 24\pi = 28.5\pi$$ - Calculate numeric value: $$28.5 \times 3.1416 \approx 89.54 \text{ cm}^2$$ --- 4. **Step 2.1.2: Total Volume of Concrete** - Volume of rectangular prism (beam): $$V_{beam} = L \times B \times h = 12 \times 8 \times 6 = 576 \text{ cm}^3$$ - Volume of one cylinder: $$V_{pillar} = \pi r^2 H = \pi (1.5)^2 (8) = \pi (2.25)(8) = 18\pi$$ - Calculate numeric value: $$18 \times 3.1416 \approx 56.55 \text{ cm}^3$$ - Volume of two cylinders: $$2 \times 56.55 = 113.10 \text{ cm}^3$$ - Total volume of concrete (beam + two pillars): $$V_{total} = 576 + 113.10 = 689.10 \text{ cm}^3$$ --- **Final Answers:** - Surface area of one closed cylinder: $\boxed{89.54 \text{ cm}^2}$. - Total volume of concrete needed: $\boxed{689.10 \text{ cm}^3}$.