Question
The three-dimensional figure is composed of three congruent rectangular prisms with the dimensions shown.
There are 3 congruent rectangular prisms. One of the rectangular prisms is 8 c-m long, 3 c-m wide and 2 c-m high.
What is the surface area, in square centimeters, of the figure?
Type your response in the box below.
There are 3 congruent rectangular prisms. One of the rectangular prisms is 8 c-m long, 3 c-m wide and 2 c-m high.
What is the surface area, in square centimeters, of the figure?
Type your response in the box below.
Answers
Answered by
GPT-4o mini
To find the surface area of the three-dimensional figure composed of three congruent rectangular prisms, we first need to calculate the surface area of one prism and then consider how adding the other prisms affects the total surface area.
The dimensions of one rectangular prism are:
- Length (l) = 8 cm
- Width (w) = 3 cm
- Height (h) = 2 cm
The formula for the surface area (SA) of a rectangular prism is given by:
\[
SA = 2(lw + lh + wh)
\]
Now, substituting the given dimensions into the formula:
\[
SA = 2(8 \times 3 + 8 \times 2 + 3 \times 2)
\]
Calculating each term inside the parentheses:
1. \(8 \times 3 = 24\)
2. \(8 \times 2 = 16\)
3. \(3 \times 2 = 6\)
Now, sum these values:
\[
24 + 16 + 6 = 46
\]
Now, multiply by 2 to find the surface area of one rectangular prism:
\[
SA = 2 \times 46 = 92 \text{ cm}^2
\]
Next, since there are three congruent prisms, the initial consideration might lead us to think that the total surface area is \(3 \times 92\). However, because some faces of these prisms will be shared between them, the total surface area will not be simply \(3 \times 92\).
To calculate the actual surface area of the figure correctly, we need to understand how the three prisms are arranged.
Assuming we place the three prisms adjacent to one another (for example, in a row), the shared faces will each reduce the outward surfaces. Each adjacent prism will share at least one face.
The effective surface area can typically be computed by determining the arrangement of the prisms and subtracting the area of the shared faces.
However, without specific details on how the prisms are arranged, we will consider them separate first:
If they were completely separate, we'd have:
\[
3 \times 92 = 276 \text{ cm}^2
\]
Yet, for a typical arrangement (like lined up), you end up reducing shared space.
Assuming they are lined up and share two whole 8 cm × 2 cm faces:
Shared area calculation:
- Shared face: \(8 \times 2 = 16\)
- There are 2 shared between 3 prisms:
Total shared area reductions:
- \(2 \times 16 = 32\) cm²
Therefore, the total surface area (without detailed layout information) while compensating for overlap conservatively could be:
Final surface area = \(276 - 32 = 244 \text{ cm}^2\)
Thus, the surface area of the figure is approximately:
\[
\boxed{244 \text{ cm}^2}
\]
Please adjust based on further layout specifics if the prisms have differing placements.
The dimensions of one rectangular prism are:
- Length (l) = 8 cm
- Width (w) = 3 cm
- Height (h) = 2 cm
The formula for the surface area (SA) of a rectangular prism is given by:
\[
SA = 2(lw + lh + wh)
\]
Now, substituting the given dimensions into the formula:
\[
SA = 2(8 \times 3 + 8 \times 2 + 3 \times 2)
\]
Calculating each term inside the parentheses:
1. \(8 \times 3 = 24\)
2. \(8 \times 2 = 16\)
3. \(3 \times 2 = 6\)
Now, sum these values:
\[
24 + 16 + 6 = 46
\]
Now, multiply by 2 to find the surface area of one rectangular prism:
\[
SA = 2 \times 46 = 92 \text{ cm}^2
\]
Next, since there are three congruent prisms, the initial consideration might lead us to think that the total surface area is \(3 \times 92\). However, because some faces of these prisms will be shared between them, the total surface area will not be simply \(3 \times 92\).
To calculate the actual surface area of the figure correctly, we need to understand how the three prisms are arranged.
Assuming we place the three prisms adjacent to one another (for example, in a row), the shared faces will each reduce the outward surfaces. Each adjacent prism will share at least one face.
The effective surface area can typically be computed by determining the arrangement of the prisms and subtracting the area of the shared faces.
However, without specific details on how the prisms are arranged, we will consider them separate first:
If they were completely separate, we'd have:
\[
3 \times 92 = 276 \text{ cm}^2
\]
Yet, for a typical arrangement (like lined up), you end up reducing shared space.
Assuming they are lined up and share two whole 8 cm × 2 cm faces:
Shared area calculation:
- Shared face: \(8 \times 2 = 16\)
- There are 2 shared between 3 prisms:
Total shared area reductions:
- \(2 \times 16 = 32\) cm²
Therefore, the total surface area (without detailed layout information) while compensating for overlap conservatively could be:
Final surface area = \(276 - 32 = 244 \text{ cm}^2\)
Thus, the surface area of the figure is approximately:
\[
\boxed{244 \text{ cm}^2}
\]
Please adjust based on further layout specifics if the prisms have differing placements.