A rectangular page is to contain 23 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.
  
Incorrect: Your answer is incorrect.
 in (smaller value)
  
Incorrect: Your answer is incorrect.
 in (larger value)

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!

A rectangular page is to contain 23 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

Discover how to minimize paper usage for a rectangular page containing 23 square inches of print with 1-inch margins. This solution utilizes optimization techniques to find dimensions approximately 7.79 inches for both width and height. Suitable for students in Grades 7-9.

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