Let's analyze the problem step by step.
Dimensions of the Blocks
Anya makes blocks from a wood piece that measures 28 inches by 28 inches by 1.5 inches. To form these blocks, she cuts the wood into 4-inch squares.
Cross Sections of a Block
When the wood is cut into blocks, the dimensions of the blocks will be:
- Length: 4 inches (from the square cut)
- Width: 4 inches (from the square cut)
- Height: 1.5 inches (thickness of the wood)
The cross sections of a block can be of two types based on how the block is sliced:
- Vertical Cross Section: This cross section would show the height of the block and the length and width.
- Horizontal Cross Section: This would show a slice from the height of the block.
Considering the Options
Identifying the options:
- Option A (1.5 in x 4 in): This could represent a vertical cross section (showing height and width).
- Option B (8 in x 8 in): This does not represent any cross section because none of the dimensions for the blocks are 8 inches.
- Option C (4 in x 4 in): This represents a horizontal cross section (showing the width and length of the block).
- Option D (1.5 in x 8 in): This does not represent any cross section since it has an irrelevant dimension (8 inches).
Correct Answers
Thus, the correct options that represent the cross sections of a block are:
- Option A (1.5 in x 4 in)
- Option C (4 in x 4 in)
Area of Each Cross Section
Now, let's calculate the area for each cross section.
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For Option A (1.5 in x 4 in): \[ \text{Area}_A = 1.5 \times 4 = 6 , \text{in}^2 \]
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For Option C (4 in x 4 in): \[ \text{Area}_C = 4 \times 4 = 16 , \text{in}^2 \]
Area of the Painted Surfaces on a Corner Piece
To find the area of the painted surfaces on a corner piece, we need to consider which surfaces are painted:
- A block has 5 painted faces: (the top, four sides)
- The bottom face is not painted.
Each face area is as follows:
- Top Face Area: \( 4 \times 4 = 16 , \text{in}^2 \)
- Each Side Face Area: \( 1.5 \times 4 = 6 , \text{in}^2 \)
Since there are 4 side faces:
- Total Area of 4 Sides: \( 4 \times 6 = 24 , \text{in}^2 \)
Total Area of Painted Surfaces
Adding the areas together: \[ \text{Total Area} = \text{Top Area} + \text{Total Side Area} = 16 , \text{in}^2 + 24 , \text{in}^2 = 40 , \text{in}^2 \]
Summary
- Cross sections of the block: Option A (1.5 in x 4 in) and Option C (4 in x 4 in)
- Areas of each cross section:
- Area of A: \(6 , \text{in}^2\)
- Area of C: \(16 , \text{in}^2\)
- Area of painted surfaces on a corner piece: 40 in².