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An illustration shows three adjoining rectangular prisms placed together and labeled from the left as 2, 1, and 3. The length of the rectangular prisms along the front base is 20 inches for each. The depth of each adjoining prism is 20 inches. The height of the rectangular prism on the left labeled 2 is 11 inches; the height of the rectangular prism in the center labeled 1 is 15 inches; the height of the rectangular prism on the right labeled 3 is 7 inches.

At the Olympics, the winners of the gold, silver, and bronze medals stand on a tiered pedestal.

As labeled in the diagram, the height of pedestal 3 is 7 inches, the height of pedestal 2 is 11 inches, and the height of pedestal 1 is 15 inches. Assuming each pedestal is of equal width and depth, find the total surface area of the stand.

(1 point)

1 answer

To find the total surface area of the tiered pedestal, we can calculate the surface area of each rectangular prism (pedestal) and then sum them up.

The formula for the surface area (SA) of a rectangular prism is given by: \[ SA = 2(lw + lh + wh) \] where:

  • \(l\) = length,
  • \(w\) = width,
  • \(h\) = height.

From the given data:

  • The length of each pedestal is 20 inches.
  • The depth (width) of each pedestal is also 20 inches.
  • The heights are:
    • Pedestal 2: 11 inches
    • Pedestal 1: 15 inches
    • Pedestal 3: 7 inches

Now we calculate the surface area for each pedestal:

  1. Pedestal 2 (height = 11 inches): \[ SA_2 = 2(20 \cdot 20 + 20 \cdot 11 + 20 \cdot 11) \] \[ = 2(400 + 220 + 220) = 2(840) = 1680 \text{ square inches} \]

  2. Pedestal 1 (height = 15 inches): \[ SA_1 = 2(20 \cdot 20 + 20 \cdot 15 + 20 \cdot 15) \] \[ = 2(400 + 300 + 300) = 2(1000) = 2000 \text{ square inches} \]

  3. Pedestal 3 (height = 7 inches): \[ SA_3 = 2(20 \cdot 20 + 20 \cdot 7 + 20 \cdot 7) \] \[ = 2(400 + 140 + 140) = 2(680) = 1360 \text{ square inches} \]

Now, we sum the surface areas of all three pedestals: \[ \text{Total Surface Area} = SA_2 + SA_1 + SA_3 \] \[ = 1680 + 2000 + 1360 = 5040 \text{ square inches} \]

Thus, the total surface area of the stand is 5040 square inches.