A rectangular page contains 100 square inches of print. The margins at the top and bottom of the page are each 1 inch deep. The margins on each side are 
1
1
2
 inches
 wide. What should the dimensions of the page be to use the least amount of paper? 

We are tasked with minimizing the total area of a rectangular page that contains 100 square inches of print, while also accounting for the margins on all sides. Let's set up the problem mathematically.

### Step 1: Define variables
- Let the width of the printed area (the part without margins) be \( w \) inches.
- Let the height of the printed area be \( h \) inches.

From the problem, we know the printed area must be 100 square inches:
\[
w \cdot h = 100
\]

### Step 2: Account for the margins
- The total width of the page includes the margins on both sides. Since the margins on each side are \( 1 \frac{1}{2} \) inches wide (i.e., 1.5 inches), the total width of the page is:
\[
W = w + 2(1.5) = w + 3
\]
- The total height of the page includes the margins at the top and bottom. Since the margins at the top and bottom are 1 inch deep each, the total height of the page is:
\[
H = h + 2(1)
= h + 2
\]

### Step 3: Minimize the total area
We aim to minimize the total area of the page, which is:
\[
A = W \cdot H = (w + 3)(h + 2)
\]
Substituting \( h = \frac{100}{w} \) from the constraint \( w \cdot h = 100 \):
\[
A(w) = (w + 3) \left( \frac{100}{w} + 2 \right)
\]

### Step 4: Simplify the expression for area
Now expand and simplify the expression for \( A(w) \):
\[
A(w) = (w + 3) \left( \frac{100}{w} + 2 \right)
= (w + 3) \left( \frac{100}{w} \right) + (w + 3)(2)
= 100 + \frac{300}{w} + 2w + 6
\]
\[
A(w) = 106 + \frac{300}{w} + 2w
\]

### Step 5: Minimize the function
To minimize \( A(w) \), take the derivative of \( A(w) \) with respect to \( w \) and set it equal to zero:
\[
A'(w) = -\frac{300}{w^2} + 2
\]
Setting \( A'(w) = 0 \):
\[
-\frac{300}{w^2} + 2 = 0
\]
\[
\frac{300}{w^2} = 2
\]
\[
w^2 = \frac{300}{2} = 150
\]
\[
w = \sqrt{150} \approx 12.247
\]

### Step 6: Find the corresponding height
Since \( w \approx 12.247 \), we can find the height using the relation \( w \cdot h = 100 \):
\[
h = \frac{100}{12.247} \approx 8.167
\]

### Step 7: Determine the dimensions of the page
The total dimensions of the page are:
- Width: \( W = w + 3 = 12.247 + 3 \approx 15.247 \) inches
- Height: \( H = h + 2 = 8.167 + 2 \approx 10.167 \) inches

Thus, the dimensions of the page should be approximately **15.25 inches by 10.17 inches** to minimize the amount of paper used.

---

Do you want details on any step, or have any questions?

Here are 5 related questions:
1. How would changing the margin size affect the dimensions?
2. Can we solve this problem using Lagrange multipliers?
3. What if the printed area was constrained to a different size, like 150 square inches?
4. How do we verify that this is a minimum and not a maximum?
5. How does this problem relate to real-life page layout design?

**Tip:** When solving optimization problems, always confirm that your critical point is a minimum (or maximum) by checking the second derivative.

A rectangular page contains 100 square inches of print. The margins at the top and bottom of the page are each 1 inch deep. The margins on each side are 1 1/2 inches wide. What should the dimensions of the page be to use the least amount of paper?

Learn how to solve an optimization problem involving a rectangular page with margins, where the goal is to minimize the total paper used. This solution involves key calculus concepts such as derivatives and critical points, suitable for students in Grades 11-12.

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