The surface area of a rectangular prism is given by the formula [1]: \[ SA_{\text{box}} = 2(lw + lh + wh) \] Substituting the values \( l = 120 \), \( w = 50 \), and \( h = 45 \) into the formula: \[ SA_{\text{box}} = 2(120 \cdot 50 + 120 \cdot 45 + 50 \cdot 45) = 2(6000 + 5400 + 2250) = 2(13,650) = 27,300 \, \text{cm}^2 \] This gives the surface area of the rectangular box as \( 27,300 \, \text{cm}^2 \) [1].

The formula for the surface area of a hemisphere is [1]: \[ SA_{\text{hemisphere}} = 2\pi r^2 \] where \( r \) is the radius. Since the diameter of the hemisphere is 50 cm, the radius is [1]: \[ r = \frac{50}{2} = 25 \, \text{cm} \] Substituting this into the formula: \[ SA_{\text{hemisphere}} = 2\pi (25)^2 = 2 \pi (625) = 1250 \pi \approx 3926.99 \, \text{cm}^2 \] Therefore, the surface area of the hemisphere lid is approximately \( 3926.99 \, \text{cm}^2 \) [1].

The total surface area is the sum of the surface area of the box and the surface area of the hemisphere lid [1]: \[ SA_{\text{total}} = SA_{\text{box}} + SA_{\text{hemisphere}} = 27,300 + 3926.99 = 31,226.99 \, \text{cm}^2 \] Therefore, the total surface area is approximately \( 31,226.99 \, \text{cm}^2 \) [1].