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 + 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.