Question

An illustration shows a three dimensional composite figure formed by a rectangular prism with a missing section. The missing section is in the shape of two rectangular prisms. With the section removed, the figure resembles three steps. The top, right, and front faces are visible. The faces and edges that are not visible are indicated by dashed lines. The rectangular prism has a length of 12 centimeters at the bottom, a width of 2 centimeters, and a height of 18 centimeters. The top length of the rectangular prism is divided equally across the three steps. Each step has a length of 4 centimeters. The right side height of the rectangular prism is divided equally along the three steps. Each step has a height of 6 centimeters.
What is the total surface area of the figure?

(1 point)
Responses

552 cm2
552 cm squared

408 cm2
408 cm squared

480 cm2
480 cm squared

444 cm2
444 cm squared

Answer 1

To calculate the total surface area of the composite figure, we need to analyze the rectangular prism's dimensions and the steps created by the missing sections.

### Given Dimensions:
- Overall length of the rectangular prism: \(l = 12 \, \text{cm}\)
- Width of the rectangular prism: \(w = 2 \, \text{cm}\)
- Height of the rectangular prism: \(h = 18 \, \text{cm}\)

### Steps Dimensions:
- Each step has a length of \(4 \, \text{cm}\).
- Each step has a height of \(6 \, \text{cm}\).
- Therefore, there are 3 steps.

### Surface Area Calculation:
A rectangular prism has surface area given by:
\[
SA = 2(lw + lh + wh)
\]

1. **Calculate Surface Area of the Entire Prism**:
\[
SA_{prism} = 2(12 \times 2 + 12 \times 18 + 2 \times 18)
\]
Calculating each term:
- \(12 \times 2 = 24\)
- \(12 \times 18 = 216\)
- \(2 \times 18 = 36\)

Now substituting into the formula:
\[
SA_{prism} = 2(24 + 216 + 36) = 2(276) = 552 \, \text{cm}^2
\]

2. **Calculate the Missing Surface Area**:
When a section is removed, the surface area of the removed parts needs to be accounted for. Each step creates additional visible surfaces:
- For each step, we will need to find the additional area for each step that was created by removing the sections.

- Each step has a footprint of \(4 \, \text{cm} \times 2 \, \text{cm}\) at the front and a vertical surface created by height \(6 \, \text{cm}\) and a length of \(4 \, \text{cm}\).

**Surface Area of Each Step**:
- Front surface: \(l \times h = 4 \, \text{cm} \times 6 \, \text{cm} = 24 \, \text{cm}^2\) (for three).
- The top of each step: \( 4 \times 2 = 8 \, \text{cm}^2\) (for three steps).
- Each vertical side for the three steps: \(2 \times 6 \times 2 = 24 \, \text{cm}^2\) (since there are two sides per step) totals to \(48 \, \text{cm}^2\) for three.

Total added area for the steps:
\[
Total \, area = 3 \times (24 + 8 + 24) = 3 \times 56 = 168 \, \text{cm}^2
\]

3. **Final Surface Area of the Figure**:
Total surface area of the modified prism:

\[
SA_{total} = SA_{prism} + area_{steps} - \text{area of missing parts}
\]
Since we don't remove anything from the visible surfaces, we retain 552 cm² total adding the new areas from the steps.

Hence, calculating the total from direct calculations and surfaces,
\[
552 \, \text{cm}^2 - total removal deductions + visible areas = 552 \, \text{cm}^2 \text{ (wouldn't count outside the prism)}
\]

### Conclusion:
The total surface area of the figure is \( \boxed{552 \, \text{cm}^2} \).