1. **Problem Statement:** Calculate the volume and surface area of a composite solid consisting of a cylinder on top of a cuboid base. 2. **Given Data:** - Cylinder radius $r = 1$ inch - Cylinder height $h_c = 4$ inches - Base is a square cuboid with side length $s = 5$ inches and height $h_b = 1$ inch 3. **Formulas:** - Volume of cylinder: $$V_c = \pi r^2 h_c$$ - Volume of cuboid: $$V_b = s^2 h_b$$ - Total volume: $$V = V_b + V_c$$ - Surface area of cylinder (excluding base on cuboid): $$A_c = 2\pi r h_c + \pi r^2$$ (lateral area plus top) - Surface area of cuboid (excluding top covered by cylinder): $$A_b = 4 s h_b + s^2 - \pi r^2$$ (all sides plus base minus area under cylinder) - Total surface area: $$A = A_b + A_c$$ 4. **Calculate volumes:** $$V_c = \pi \times 1^2 \times 4 = 4\pi$$ $$V_b = 5^2 \times 1 = 25$$ $$V = 25 + 4\pi$$ 5. **Calculate surface areas:** - Lateral surface area of cuboid: $$4 \times 5 \times 1 = 20$$ - Base area of cuboid: $$5^2 = 25$$ - Area under cylinder on base: $$\pi \times 1^2 = \pi$$ - Adjusted base area: $$25 - \pi$$ - Cylinder lateral surface area: $$2 \pi \times 1 \times 4 = 8\pi$$ - Cylinder top area: $$\pi \times 1^2 = \pi$$ - Total surface area: $$A = 20 + (25 - \pi) + 8\pi + \pi = 20 + 25 - \pi + 8\pi + \pi = 45 + 8\pi$$ 6. **Final answers:** - Volume: $$V = 25 + 4\pi \approx 25 + 12.57 = 37.57 \text{ cubic inches}$$ - Surface area: $$A = 45 + 8\pi \approx 45 + 25.13 = 70.13 \text{ square inches}$$ These calculations give the total volume and surface area of the composite solid.