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 15 inches, a width of 6 inches, and a height of 12 inches. The missing section is removed from the top center of the rectangular prism. The missing section is a rectangular prism with a length of 7 inches, and a height of 6 inches. The missing section has the same width as the larger rectangular prism. The missing section is centered at the top of the larger rectangular prism. The remaining lengths to the left and right of the missing section both measure 4 inches.
What is the total surface area of the figure?
(1 point)
in.2
Remaining Attempts : 3
Skip to navigation
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 15 inches, a width of 6 inches, and a height of 12 inches. The missing section is removed from the top center of the rectangular prism. The missing section is a rectangular prism with a length of 7 inches, and a height of 6 inches. The missing section has the same width as the larger rectangular prism. The missing section is centered at the top of the larger rectangular prism. The remaining lengths to the left and right of the missing section both measure 4 inches.
What is the total surface area of the figure?
(1 point)
in.2
Remaining Attempts : 3
Skip to navigation
Answers
Answered by
GPT-4o mini
To find the total surface area of the composite figure, we need to calculate the surface area of the larger rectangular prism, subtract the surface area of the missing section, and then add the area of the new exposed faces created by the removal of the section.
### Step 1: Surface Area of the Larger Rectangular Prism
The formula for the surface area \( S \) of a rectangular prism is:
\[
S = 2(lw + lh + wh)
\]
Where:
- \( l \) is the length
- \( w \) is the width
- \( h \) is the height
For the larger rectangular prism:
- Length \( l = 15 \) inches
- Width \( w = 6 \) inches
- Height \( h = 12 \) inches
Calculating the surface area:
\[
S = 2(15 \times 6 + 15 \times 12 + 6 \times 12)
\]
\[
= 2(90 + 180 + 72)
\]
\[
= 2(342)
\]
\[
= 684 \text{ in}^2
\]
### Step 2: Surface Area of the Missing Section
For the missing section (also a rectangular prism):
- Length \( l = 7 \) inches
- Width \( w = 6 \) inches
- Height \( h = 6 \) inches
Calculating the surface area of the missing section:
\[
S = 2(7 \times 6 + 7 \times 6 + 6 \times 6)
\]
\[
= 2(42 + 42 + 36)
\]
\[
= 2(120)
\]
\[
= 240 \text{ in}^2
\]
### Step 3: Accounting for New Exposed Faces
When the section is removed, we have new exposed faces:
1. The bottom face of the missing section (which is \( 7 \times 6 = 42 \) square inches).
2. The front face of the missing section (which is \( 7 \times 6 = 42 \) square inches).
3. The back face of the missing section (which is also \( 7 \times 6 = 42 \) square inches).
Total area of the new exposed faces:
\[
42 + 42 + 42 = 126 \text{ in}^2
\]
### Step 4: Final Calculation of Total Surface Area
Now, combining all these together:
\[
\text{Total Surface Area} = \text{Surface Area of Larger Prism} - \text{Surface Area of Missing Section} + \text{Area of New Exposed Faces}
\]
\[
= 684 - 240 + 126
\]
\[
= 684 - 240 + 126 = 570 \text{ in}^2
\]
Thus, the total surface area of the composite figure is:
\[
\boxed{570} \text{ in}^2
\]
### Step 1: Surface Area of the Larger Rectangular Prism
The formula for the surface area \( S \) of a rectangular prism is:
\[
S = 2(lw + lh + wh)
\]
Where:
- \( l \) is the length
- \( w \) is the width
- \( h \) is the height
For the larger rectangular prism:
- Length \( l = 15 \) inches
- Width \( w = 6 \) inches
- Height \( h = 12 \) inches
Calculating the surface area:
\[
S = 2(15 \times 6 + 15 \times 12 + 6 \times 12)
\]
\[
= 2(90 + 180 + 72)
\]
\[
= 2(342)
\]
\[
= 684 \text{ in}^2
\]
### Step 2: Surface Area of the Missing Section
For the missing section (also a rectangular prism):
- Length \( l = 7 \) inches
- Width \( w = 6 \) inches
- Height \( h = 6 \) inches
Calculating the surface area of the missing section:
\[
S = 2(7 \times 6 + 7 \times 6 + 6 \times 6)
\]
\[
= 2(42 + 42 + 36)
\]
\[
= 2(120)
\]
\[
= 240 \text{ in}^2
\]
### Step 3: Accounting for New Exposed Faces
When the section is removed, we have new exposed faces:
1. The bottom face of the missing section (which is \( 7 \times 6 = 42 \) square inches).
2. The front face of the missing section (which is \( 7 \times 6 = 42 \) square inches).
3. The back face of the missing section (which is also \( 7 \times 6 = 42 \) square inches).
Total area of the new exposed faces:
\[
42 + 42 + 42 = 126 \text{ in}^2
\]
### Step 4: Final Calculation of Total Surface Area
Now, combining all these together:
\[
\text{Total Surface Area} = \text{Surface Area of Larger Prism} - \text{Surface Area of Missing Section} + \text{Area of New Exposed Faces}
\]
\[
= 684 - 240 + 126
\]
\[
= 684 - 240 + 126 = 570 \text{ in}^2
\]
Thus, the total surface area of the composite figure is:
\[
\boxed{570} \text{ in}^2
\]
There are no human answers yet. A form for humans to post answers is coming very soon!