A square prism measuring 2 m by 2 m in base length and 5 m in height has a half-sphere attached to the top. What is the total surface area of this composite solid?

To find the total surface area of the composite solid, we need to calculate the surface areas of the square prism and the half-sphere separately, then combine them. 1. **Surface Area of the Square Prism:** The square prism has a base that is a square with side length \( 2 \, \text{m} \) and a height of \( 5 \, \text{m} \). - Area of the base (bottom): \[ \text{Base Area} = 2 \times 2 = 4 \, \text{m}^2 \] - Area of the sides (4 rectangles): \[ \text{Lateral Area} = 4 \times (\text{base length} \times \text{height}) = 4 \times (2 \times 5) = 40 \, \text{m}^2 \] - Total surface area of the prism (excluding the top base): \[ \text{Total Prism Area} = \text{Base Area} + \text{Lateral Area} = 4 + 40 = 44 \, \text{m}^2 \] 2. **Surface Area of the Half-Sphere:** The radius of the half-sphere is half of the base length of the prism: \[ r = \frac{2}{2} = 1 \, \text{m} \] - Surface area of a full sphere: \[ \text{Sphere Area} = 4\pi r^2 = 4\pi (1)^2 = 4\pi \, \text{m}^2 \] - Since we only have a half-sphere: \[ \text{Half-Sphere Area} = 2\pi r^2 = 2\pi (1)^2 = 2\pi \, \text{m}^2 \] 3. **Total Surface Area of the Composite Solid:** Since the top base of the prism is covered by the half-sphere, we do not include it in the calculations for the total surface area: \[ \text{Total Surface Area} = \text{Total Prism Area} + \text{Half-Sphere Area} = 44 + 2\pi \] Calculating \(2\pi\) (approximately \(6.28\)): \[ \text{Total Surface Area} \approx 44 + 6.28 \approx 50.28 \, \text{m}^2 \] Thus, the total surface area of the composite solid is approximately: \[ \text{Total Surface Area} \approx 50.28 \, \text{m}^2 \]