33. A board $$ 1 \frac{5}{8} $$ inches by $$ 3 \frac{5}{8} $$ inches is $$ 6 \frac{1}{2} $$ feet long. You want to cut as many pieces as possible from the board so that each piece is $$ 1 \frac{5}{8} $$ inches by $$ 3 \frac{5}{8} $$ inches and 6 inches long. Each saw cut wastes $$ \frac{1}{8} $$ inch of the board. How many 6 -inch-long pieces will you be able to cut?

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To determine how many 6-inch-long pieces can be cut from a board, we'll break down the problem step by step.

### Given:
- **Board Dimensions**: $$1 \frac{5}{8}$$ inches by $$3 \frac{5}{8}$$ inches.
- **Board Length**: $$6 \frac{1}{2}$$ feet (which is $$6 \times 12 + 6 = 78$$ inches).
- **Piece Dimensions**: Each piece is $$1 \frac{5}{8}$$ inches by $$3 \frac{5}{8}$$ inches by 6 inches.
- **Waste per Cut**: $$\frac{1}{8}$$ inch.

### Steps to Solve:

1. **Calculate the Total Length Available**:
   - The board is 78 inches long.

2. **Determine Length Used by Each Piece and Waste**:
   - Each 6-inch piece requires exactly 6 inches of the board.
   - Each cut results in a waste of $$\frac{1}{8}$$ inch.

3. **Set Up the Inequality**:
   - Let $$n$$ be the number of 6-inch pieces.
   - Total length used by pieces: $$6n$$ inches.
   - Total waste from cuts: $$(n - 1) \times \frac{1}{8}$$ inches (since there's no waste after the last piece).
   - The inequality representing the total length used must be less than or equal to 78 inches:
     $$
     6n + \frac{n - 1}{8} \leq 78
     $$

4. **Solve the Inequality**:
   - Multiply everything by 8 to eliminate the fraction:
     $$
     48n + (n - 1) \leq 624
     $$
     $$
     49n - 1 \leq 624
     $$
     $$
     49n \leq 625
     $$
     $$
     n \leq \frac{625}{49} \approx 12.755
     $$
   - Since you can't have a fraction of a piece, the maximum number of whole pieces you can cut is **12**.

5. **Verification**:
   - For 12 pieces:
     - Total length used: $$12 \times 6 = 72$$ inches.
     - Total waste: $$11 \times \frac{1}{8} = 1.375$$ inches.
     - Combined: $$72 + 1.375 = 73.375$$ inches, which is within the 78-inch limit.
   - For 13 pieces:
     - Total length used: $$13 \times 6 = 78$$ inches.
     - Total waste: $$12 \times \frac{1}{8} = 1.5$$ inches.
     - Combined: $$78 + 1.5 = 79.5$$ inches, which exceeds the 78-inch limit.

### **Answer**:  
12 six-inch
You can cut **12** six-inch-long pieces from the board.

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