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.