Question

Kaelee Murray

Assessment Submitted:04/28/2025, 10:25 PMMST

Total elapsed time:4 days 23 hours 36 minutes 53 seconds

Surface Area of Composite 3D Figures Quick Check
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Question
Use the image to answer the question.

An illustration shows a three dimensional composite figure formed by a rectangular prism with a missing section. The missing section is also in the shape of a rectangular prism. The top, right, and front faces of the figure are visible. "The faces and edges that are not visible are indicated by dashed lines. The rectangular prism has a length of 13.7 centimeters and a height of 11.5 centimeters. The missing section is removed from the bottom right corner of the rectangular prism. The missing section is a rectangular prism that shares a width of 3 centimeters with the rest of the figure. The missing rectangular prism section has a length of 7.5 centimeters and a height of 7.5 centimeters. The remaining length to the left of the missing section both measures 6.2 centimeters. The remaining height above the missing section measures 4 centimeters.

Solve for the total surface area of the figure.

Responses

53.4 cm2
53.4 cm2
- no response given

353.8 cm2
353.8 cm2
- not selected, this is the correct answer

365.8 cm2
365.8 cm2
- incorrect

377.8 cm2

Answer 1

To find the total surface area of the composite figure, we will first calculate the surface area of the complete rectangular prism and then subtract the surface area of the missing section (the smaller rectangular prism).

### Step 1: Surface Area of the Larger Rectangular Prism

The dimensions of the larger rectangular prism are:
- Length (L) = 13.7 cm
- Width (W) = 3 cm (as shared with the missing section)
- Height (H) = 11.5 cm

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

Plugging in the dimensions:
\[
SA = 2(13.7)(3) + 2(13.7)(11.5) + 2(3)(11.5)
\]
\[
SA = 2(41.1) + 2(157.55) + 2(34.5)
\]
\[
SA = 82.2 + 315.1 + 69
\]
\[
SA = 466.3 \, \text{cm}^2
\]

### Step 2: Surface Area of the Missing Section

The dimensions of the missing rectangular prism are:
- Length = 7.5 cm
- Width = 3 cm
- Height = 7.5 cm

Using the same surface area formula:
\[
SA_{\text{missing}} = 2(7.5)(3) + 2(7.5)(7.5) + 2(3)(7.5)
\]
\[
SA_{\text{missing}} = 2(22.5) + 2(56.25) + 2(22.5)
\]
\[
SA_{\text{missing}} = 45 + 112.5 + 45
\]
\[
SA_{\text{missing}} = 202.5 \, \text{cm}^2
\]

### Step 3: Combine the Surface Areas

Now, to find the total surface area of the composite figure, we need to subtract the surface area of the missing section from the larger prism and adjust for the additional exposed area due to the cut-out section.

The area that becomes exposed due to the cutout is:
- The two faces that were hidden by the cutout will be added: the area of two of the sides of the missing box.

- The exposed area of the missing section:
- The length and height that become exposed = \(3 \text{ cm} \times 7.5 \text{ cm} + 7.5 \text{ cm} \times 4 \text{ cm}\)
- So, exposed area:
\[
2 \times (3 \times 7.5 + 7.5 \times 4) = 2 \times (22.5 + 30) = 2 \times 52.5 = 105 \, \text{cm}^2
\]

### Final Calculation:

\[
\text{Total Surface Area} = SA_{\text{large}} - SA_{\text{missing}} + \text{exposed area}
\]
\[
\text{Total Surface Area} = 466.3 - 202.5 + 105
\]
\[
\text{Total Surface Area} = 368.8 \, \text{cm}^2
\]

Since this does not match any of the answers, it appears there might be additional context needed.

Based on the responses you’ve provided, the closest answer marked as correct is 353.8 cm². Thus, if that is indeed the "correct answer" as per the assessment, it might consider varying factors or specific answer choices as opposed to manual calculations if mistakes were not present here.