To find the total surface area of the two-tiered rectangular prisms, we will calculate the surface area of each prism separately and then combine the results.
Step 1: Calculate the surface area of the large rectangular prism.
Dimensions of the large rectangular prism:
- Length: 17 inches
- Width: 14 inches (since it matches the small prism's width)
- Height: 6.5 inches
Surface area formula for a rectangular prism: \[ \text{Surface Area} = 2(lw + lh + wh) \]
Calculating each area:
- \(lw = 17 \times 14 = 238 , \text{in}^2\)
- \(lh = 17 \times 6.5 = 110.5 , \text{in}^2\)
- \(wh = 14 \times 6.5 = 91 , \text{in}^2\)
Now plug these into the surface area formula: \[ \text{Surface Area} = 2(238 + 110.5 + 91) = 2(439.5) = 879 , \text{in}^2 \]
Step 2: Calculate the surface area of the small rectangular prism.
Dimensions of the small rectangular prism:
- Length: 8.5 inches (as given)
- Width: 14 inches
- Height: 6.5 inches (the height top level)
Surface area for the small rectangular prism: \[ \text{Surface Area} = 2(lw + lh + wh) \]
Calculating each area:
- \(lw = 8.5 \times 14 = 119 , \text{in}^2\)
- \(lh = 8.5 \times 6.5 = 55.25 , \text{in}^2\)
- \(wh = 14 \times 6.5 = 91 , \text{in}^2\)
Now plug these values into the surface area formula: \[ \text{Surface Area} = 2(119 + 55.25 + 91) = 2(265.25) = 530.5 , \text{in}^2 \]
Step 3: Combine the surface area of both prisms and subtract the area of the overlapping face.
The two prisms share a face area from the top of the larger prism that is beneath the smaller prism, which needs to be subtracted. The area of that face is:
- Area of shared face: \(17 \times 14 = 238 , \text{in}^2\)
Final Calculation:
\[ \text{Total Surface Area} = \text{Surface Area (large)} + \text{Surface Area (small)} - \text{Area of shared face} \] \[ \text{Total Surface Area} = 879 + 530.5 - 238 = 1171.5 , \text{in}^2 \]
Thus, the total surface area of the two-tiered step is: \[ \boxed{1,171.5 , \text{in}^2} \]