Question

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)

Answer 1

To find the total surface area of the stand composed of three adjoining rectangular prisms (pedestals), we need to calculate the surface area of each individual prism and then combine them.

### Dimensions of the prisms:
1. Prism 2: Height = 11 inches, Width = 20 inches, Depth = 20 inches
2. Prism 1: Height = 15 inches, Width = 20 inches, Depth = 20 inches
3. Prism 3: Height = 7 inches, Width = 20 inches, Depth = 20 inches

### Surface Area of a Rectangular Prism Formula:
The surface area \( S \) of a rectangular prism is given by:
\[
S = 2lw + 2lh + 2wh
\]
where:
- \( l \) = length (20 inches)
- \( w \) = width (20 inches)
- \( h \) = height (varies for each prism)

### Calculations:

1. **Calculating Surface Area of Prism 2:**
- Height \( h = 11 \) inches
\[
S_2 = 2(20)(20) + 2(20)(11) + 2(20)(11) = 800 + 440 + 440 = 1680 \text{ square inches}
\]

2. **Calculating Surface Area of Prism 1:**
- Height \( h = 15 \) inches
\[
S_1 = 2(20)(20) + 2(20)(15) + 2(20)(15) = 800 + 600 + 600 = 2000 \text{ square inches}
\]

3. **Calculating Surface Area of Prism 3:**
- Height \( h = 7 \) inches
\[
S_3 = 2(20)(20) + 2(20)(7) + 2(20)(7) = 800 + 280 + 280 = 1360 \text{ square inches}
\]

### Total Surface Area:
Now, we need to combine the surface areas of each prism. However, since these prisms are adjoining, the areas that connect the prisms will not be counted twice. Each adjoining prism shares a face of area \( 20 \times 20 = 400 \) square inches where they touch.

- **Shared Face between Prism 2 and Prism 1:** 400 square inches
- **Shared Face between Prism 1 and Prism 3:** 400 square inches

So, the adjustment for the shared areas is:
- Total shared area = \( 400 + 400 = 800 \) square inches

Now, we find the total surface area:
\[
\text{Total Surface Area} = S_2 + S_1 + S_3 - \text{Shared Area} = 1680 + 2000 + 1360 - 800 = 4240 \text{ square inches}
\]

### Conclusion:
The total surface area of the stand is **4240 square inches**.