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.
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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).
- Each new prism will have:
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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:
\[ \text{Surface Area of one prism} = 27 , \text{cm}^2 + 9 , \text{cm}^2 = 36 , \text{cm}^2 \]
- 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
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