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An illustration shows a small rectangular prism placed on top of a large rectangular prism, to form a 2-tiered step. The edges that are not visible are marked with dashed lines. The length of the large rectangular prism on the bottom is 17 inches and the height is 6.5 inches. The small rectangular prism placed on top along the back edge shows a width of 14 inches, which aligns with the width of the bottom rectangular prism. The total height of both rectangles is 13 inches overall. The length of the large rectangular prism that is not covered by the small rectangular prism is 8.5 inches.

Find the surface area of the two-tiered steps.

(1 point)
Responses

1,171.5 in.2
1,171.5 in. squared

879 in.2
879 in. squared

1,409.5 in.2
1,409.5 in. squared

2,320.5 in.2
2,320.5 in. squared
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1 answer

To find the surface area of the two-tiered rectangular prisms, we need to calculate the surface area of both prisms and then consider the overlapping areas where they connect.

  1. Surface Area of the Large Rectangular Prism:

    • Dimensions: Length = 17 inches, Width = 14 inches, Height = 6.5 inches.
    • Surface Area = 2lw + 2lh + 2wh
    • Calculate:
      • lw = 17 * 14 = 238
      • lh = 17 * 6.5 = 110.5
      • wh = 14 * 6.5 = 91
    • Surface Area = 2(238) + 2(110.5) + 2(91) = 476 + 221 + 182 = 879 in²

  2. Surface Area of the Small Rectangular Prism:

    • Dimensions: Width = 14 inches, Length (uncovered part of large prism) + covered part = 17 - 8.5 = 8.5 inches. Total height (with the large prism) = 13 - 6.5 = 6.5 inches.
    • Surface Area = 2lw + 2lh + 2wh
    • Surface Area = 2(8.5)(14) + 2(8.5)(6.5) + 2(14)(6.5)
    • Calculate:
      • lw = 8.5 * 14 = 119
      • lh = 8.5 * 6.5 = 55.25
      • wh = 14 * 6.5 = 91
    • Surface Area = 2(119) + 2(55.25) + 2(91) = 238 + 110.5 + 182 = 530 in²

  3. Calculate the total surface area:

    • Total Surface Area = Surface area of large prism + Surface area of small prism
    • Surface Area = 879 + 530 = 1409 in²

Thus, the surface area of the two-tiered steps is 1,409.5 in².