Question
The area (A) of a rectangular
surface is the product of its
length and its width.
A = l × w
Area has square units, such
as square centimeters (cm2
).
The surface area (SA) of a
rectangular solid is the sum
of the areas of all of its sides.
3. Use Geometry A
rock sample is a cube and
measures 3 cm on each side.
a. What is the surface area of the rock If you break the sample
into two equal parts,
what is the total surface
area now?
surface is the product of its
length and its width.
A = l × w
Area has square units, such
as square centimeters (cm2
).
The surface area (SA) of a
rectangular solid is the sum
of the areas of all of its sides.
3. Use Geometry A
rock sample is a cube and
measures 3 cm on each side.
a. What is the surface area of the rock If you break the sample
into two equal parts,
what is the total surface
area now?
Answers
GPT-4o mini
To find the surface area of a cube, you can use the formula:
\[
\text{SA} = 6 \times l^2
\]
where \( l \) is the length of one side of the cube.
Given that each side of the rock sample is 3 cm:
\[
\text{SA} = 6 \times (3 \, \text{cm})^2 = 6 \times 9 \, \text{cm}^2 = 54 \, \text{cm}^2
\]
So, the surface area of the rock sample is 54 cm².
### If you break the sample into two equal parts:
When you break a cube into two equal parts, you make a cut through one dimension. This results in two rectangular prisms. Each of these prisms will have 3 faces from the original cube (that remain the same) and 2 additional faces from the cut.
1. **Calculating the new surface area per prism**:
- Each new prism will have:
- 3 original faces from the cube, each of which measures \( 3 \, \text{cm} \times 3 \, \text{cm} \).
- 2 new faces, which are created from the cut. Since the cut divides the cube into two equal halves, these faces will be \( 3 \, \text{cm} \times 1.5 \, \text{cm} \) (the height of each prism will be half the original height).
2. **Surface area of one prism**:
- Original faces: \( 3 \times (3 \, \text{cm} \times 3 \, \text{cm}) = 3 \times 9 \, \text{cm}^2 = 27 \, \text{cm}^2 \)
- New faces from the cut: \( 2 \times (3 \, \text{cm} \times 1.5 \, \text{cm}) = 2 \times 4.5 \, \text{cm}^2 = 9 \, \text{cm}^2 \)
Adding these together gives the surface area for one prism:
\[
\text{Surface Area of one prism} = 27 \, \text{cm}^2 + 9 \, \text{cm}^2 = 36 \, \text{cm}^2
\]
3. **Total surface area for the two prisms**:
Multiply the surface area of one prism by 2:
\[
\text{Total Surface Area} = 2 \times 36 \, \text{cm}^2 = 72 \, \text{cm}^2
\]
### Summary:
- **Original surface area of the cube**: 54 cm²
- **Total surface area after breaking into two equal parts**: 72 cm²
\[
\text{SA} = 6 \times l^2
\]
where \( l \) is the length of one side of the cube.
Given that each side of the rock sample is 3 cm:
\[
\text{SA} = 6 \times (3 \, \text{cm})^2 = 6 \times 9 \, \text{cm}^2 = 54 \, \text{cm}^2
\]
So, the surface area of the rock sample is 54 cm².
### If you break the sample into two equal parts:
When you break a cube into two equal parts, you make a cut through one dimension. This results in two rectangular prisms. Each of these prisms will have 3 faces from the original cube (that remain the same) and 2 additional faces from the cut.
1. **Calculating the new surface area per prism**:
- Each new prism will have:
- 3 original faces from the cube, each of which measures \( 3 \, \text{cm} \times 3 \, \text{cm} \).
- 2 new faces, which are created from the cut. Since the cut divides the cube into two equal halves, these faces will be \( 3 \, \text{cm} \times 1.5 \, \text{cm} \) (the height of each prism will be half the original height).
2. **Surface area of one prism**:
- Original faces: \( 3 \times (3 \, \text{cm} \times 3 \, \text{cm}) = 3 \times 9 \, \text{cm}^2 = 27 \, \text{cm}^2 \)
- New faces from the cut: \( 2 \times (3 \, \text{cm} \times 1.5 \, \text{cm}) = 2 \times 4.5 \, \text{cm}^2 = 9 \, \text{cm}^2 \)
Adding these together gives the surface area for one prism:
\[
\text{Surface Area of one prism} = 27 \, \text{cm}^2 + 9 \, \text{cm}^2 = 36 \, \text{cm}^2
\]
3. **Total surface area for the two prisms**:
Multiply the surface area of one prism by 2:
\[
\text{Total Surface Area} = 2 \times 36 \, \text{cm}^2 = 72 \, \text{cm}^2
\]
### Summary:
- **Original surface area of the cube**: 54 cm²
- **Total surface area after breaking into two equal parts**: 72 cm²