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The problem is asking to minimize the area of a rectangular page, which contains 30 square inches of print, with 1-inch margins on all sides.

### Let’s define the variables:
- Let the dimensions of the printed area (excluding margins) be \( x \times y \), where:
  - \( x \) is the width of the printed area.
  - \( y \) is the height of the printed area.
  
- The total area of the printed section is given as:
  \[
  x \cdot y = 30 \quad \text{(since the printed area is 30 square inches)}
  \]

- The total dimensions of the entire page will include the margins:
  - The width of the entire page is \( x + 2 \) (since there’s 1-inch margin on both sides).
  - The height of the entire page is \( y + 2 \) (since there’s 1-inch margin on the top and bottom).

- The total area \( A \) of the page is:
  \[
  A = (x + 2)(y + 2)
  \]

Now, we need to express this total area \( A \) in terms of a single variable and find the minimum.

### Step 1: Express \( y \) in terms of \( x \)
From the equation \( x \cdot y = 30 \), solve for \( y \):
\[
y = \frac{30}{x}
\]

### Step 2: Substitute \( y \) into the area formula
Substitute \( y = \frac{30}{x} \) into the total area equation \( A \):
\[
A = (x + 2) \left( \frac{30}{x} + 2 \right)
\]
Simplify this expression:
\[
A = (x + 2) \left( \frac{30}{x} + 2 \right)
   = (x + 2) \left( \frac{30 + 2x}{x} \right)
   = \frac{(x + 2)(30 + 2x)}{x}
\]

### Step 3: Differentiate \( A \) with respect to \( x \)
To find the minimum area, we take the derivative of \( A \) with respect to \( x \) and set it equal to zero. Let’s calculate this derivative.

Let me calculate that now.The critical points found are \( x = \sqrt{30} \) and \( x = -\sqrt{30} \). Since \( x \) represents a dimension of the printed area, it must be positive, so we have:

\[
x = \sqrt{30} \approx 5.477 \, \text{inches}
\]

### Step 4: Calculate \( y \)
Using the relationship \( y = \frac{30}{x} \), we can now find \( y \):

\[
y = \frac{30}{\sqrt{30}} = \sqrt{30} \approx 5.477 \, \text{inches}
\]

### Step 5: Total dimensions of the page
- The total width of the page is \( x + 2 = \sqrt{30} + 2 \approx 7.477 \) inches.
- The total height of the page is \( y + 2 = \sqrt{30} + 2 \approx 7.477 \) inches.

Thus, the dimensions of the page that minimize the area while containing 30 square inches of print with 1-inch margins on all sides are approximately:
\[
7.477 \times 7.477 \, \text{inches}.
\]

Would you like a detailed breakdown of any step, or do you have any other questions?

Here are 5 related questions to explore:
1. How does changing the margin size affect the dimensions of the page?
2. What happens to the minimum area if the printed area is changed to 40 square inches?
3. Can we generalize this method for a page with unequal margins?
4. How can we verify that this critical point yields a minimum?
5. How do we calculate the second derivative to confirm a minimum?

**Tip:** Always check the second derivative to confirm whether a critical point is a minimum or maximum in optimization problems.

A rectangular page is to contain 30 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.

Solve the problem of minimizing the area of a rectangular page containing 30 square inches of print with 1-inch margins on each side. Learn how to apply calculus and algebra to find the optimal page dimensions, with a step-by-step guide. Suitable for advanced high school students.

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