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To solve this problem, we aim to minimize the amount of paper used for a rectangular document while ensuring that the printed area is 64 square inches. Let’s define the problem mathematically:

### Definitions:
1. Let the **printed area** have dimensions \( x \) (width) and \( y \) (height), so \( x \cdot y = 64 \) square inches.
2. Since there is a 0.5-inch margin on all sides, the **total dimensions** of the document will be:
   - Total width = \( x + 1 \) (adding 0.5-inch margins on both sides).
   - Total height = \( y + 1 \) (adding 0.5-inch margins on both sides).

The **total paper area** to minimize is:
\[
A = (x + 1)(y + 1).
\]

### Substitution:
From the constraint \( x \cdot y = 64 \), solve for \( y \) in terms of \( x \):
\[
y = \frac{64}{x}.
\]

Substitute \( y \) into the area equation:
\[
A = (x + 1)\left(\frac{64}{x} + 1\right).
\]

Simplify:
\[
A = x \cdot \frac{64}{x} + x + \frac{64}{x} + 1 = 64 + x + \frac{64}{x} + 1.
\]
\[
A = x + \frac{64}{x} + 65.
\]

### Minimizing \( A \):
To minimize \( A \), take the derivative with respect to \( x \) and set it to 0:
\[
\frac{dA}{dx} = 1 - \frac{64}{x^2}.
\]
Set \( \frac{dA}{dx} = 0 \):
\[
1 - \frac{64}{x^2} = 0 \implies \frac{64}{x^2} = 1 \implies x^2 = 64 \implies x = 8 \; (\text{since dimensions are positive}).
\]

### Solve for \( y \):
Using \( x \cdot y = 64 \):
\[
y = \frac{64}{8} = 8.
\]

### Dimensions of the document:
- Printed area dimensions: \( x = 8 \), \( y = 8 \).
- Total document dimensions (including margins): 
  - Width = \( x + 1 = 8 + 1 = 9 \),
  - Height = \( y + 1 = 8 + 1 = 9 \).

Thus, the dimensions of the document that minimize paper usage are **9 inches by 9 inches**.

---

Would you like more details about the steps? Here are 5 related questions you might find helpful:
1. What if the margin size was different (e.g., 1 inch)?
2. How can this method be generalized for other printed areas?
3. What role does calculus play in optimization problems like this?
4. Can we use a numerical method to verify the result?
5. How would the solution change if the paper dimensions must be integers?

**Tip:** When solving optimization problems, always write down the constraints and objective clearly to avoid confusion.

A rectangular document is to contain 64 square inches of print with a margin of 0.5 inch on each side. Find the dimensions of the document such that the least amount of paper is used.

Learn how to find the dimensions of a rectangular document with 64 square inches of print and a 0.5-inch margin, minimizing paper usage. Explore concepts of optimization, algebra, and differentiation with a step-by-step solution suitable for Grades 10-12.

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