To find the dimensions of the page that minimize the total area while containing 23 square inches of print, we can set up the problem as follows:

1. Let $x$ be the width of the printed area, and $y$ be the height of the printed area.
2. Given that $xy = 23$ (the area of print).
3. The total dimensions of the page, including margins, will be $(x + 2)$ and $(y + 2)$ because of the 1-inch margins on each side.

The total area $A$ of the page can be expressed as: $A = (x + 2)(y + 2) = xy + 2x + 2y + 4$ Substituting $y = \frac{23}{x}$: $A = 23 + 2x + 2\left(\frac{23}{x}\right) + 4 = 27 + 2x + \frac{46}{x}$

To minimize $A$, we can take the derivative and set it to zero: $\frac{dA}{dx} = 2 - \frac{46}{x^2}$ Setting the derivative equal to zero: $2 - \frac{46}{x^2} = 0 \implies 2x^2 = 46 \implies x^2 = 23 \implies x = \sqrt{23}$

Now, calculate $y$: $y = \frac{23}{\sqrt{23}} = \sqrt{23}$

The dimensions of the page are:

• Width: $x + 2 = \sqrt{23} + 2$
• Height: $y + 2 = \sqrt{23} + 2$

So, the dimensions of the page that minimize the paper used are approximately:

• Width: $\sqrt{23} + 2 \approx 7.79$
• Height: $\sqrt{23} + 2 \approx 7.79$

Would you like more details on the calculations, or do you have any questions? Here are some related questions to consider:

1. How does changing the area of print affect the dimensions of the page?
2. What would happen if the margins were increased to 2 inches?
3. Can you apply this optimization technique to other shapes or constraints?
4. How can you verify that the calculated dimensions minimize the area?
5. What real-world factors might influence the dimensions of printed pages beyond just area?

Tip: Always check the second derivative to confirm that you have found a minimum point!