We are tasked with finding the dimensions of a rectangular page that minimizes the amount of paper used, while still containing 162 square inches of printed area. The margins on the page are as follows:

• 2 inches for the top and bottom.
• 1 inch for each side.

Step 1: Define Variables

• Let $x$ represent the width of the page.
• Let $y$ represent the height of the page.

Step 2: Express the printed area in terms of $x$ and $y$

The printed area is 162 square inches, and this is the area that excludes the margins. The width of the printed area is $x - 2$ because the margins on the sides are 1 inch each (total 2 inches), and the height of the printed area is $y - 4$ because the top and bottom margins are 2 inches each (total 4 inches).

Thus, the printed area is given by:

$(x - 2)(y - 4) = 162$

Step 3: Find the total area of the paper

The total area of the paper is the entire page, including the margins. This area is $x \times y$.

Step 4: Minimize the total area of the paper

To minimize the total paper area, we first need to express $y$ in terms of $x$ from the printed area equation:

$(x - 2)(y - 4) = 162$

Solve for $y$:

$y - 4 = \frac{162}{x - 2}$

$y = \frac{162}{x - 2} + 4$

Step 5: Express the total area in terms of $x$

Now, the total area of the paper is:

$A = x \times y = x \times \left( \frac{162}{x - 2} + 4 \right)$

Simplify this expression for the area:

$A = x \times \frac{162}{x - 2} + 4x$

Step 6: Take the derivative of $A$ with respect to $x$

To minimize the area, we differentiate $A$ with respect to $x$. Use the quotient rule for the first term:

$\frac{d}{dx}\left( x \times \frac{162}{x - 2} \right) = \frac{162(x - 2) - 162x}{(x - 2)^2} = \frac{-324}{(x - 2)^2}$

Thus, the derivative of $A$ is:

$\frac{dA}{dx} = \frac{-324}{(x - 2)^2} + 4$

Step 7: Set the derivative equal to zero to find the critical points

Set $\frac{dA}{dx} = 0$:

$\frac{-324}{(x - 2)^2} + 4 = 0$

Solving for $x$:

$\frac{-324}{(x - 2)^2} = -4$

$\frac{324}{(x - 2)^2} = 4$

$(x - 2)^2 = \frac{324}{4} = 81$

$x - 2 = 9 \quad \text{or} \quad x - 2 = -9$

Thus, $x = 11$ or $x = -7$.

Since $x$ must be positive, we take $x = 11$.

Step 8: Solve for $y$

Substitute $x = 11$ into the equation for $y$:

$y = \frac{162}{x - 2} + 4 = \frac{162}{11 - 2} + 4 = \frac{162}{9} + 4 = 18 + 4 = 22$

Final Answer

The dimensions of the page that minimize the amount of paper used are:

• $x = 11$ inches (width)
• $y = 22$ inches (height)

Do you have any questions or need further details? Here are a few related topics you might want to explore:

1. What is the concept of optimization in calculus?
2. How do you apply the derivative to minimize or maximize functions?
3. What is the significance of the second derivative test in optimization problems?
4. Can this method be applied to other real-world optimization problems?
5. How do you calculate the total area of a rectangle with given constraints?

Tip: When solving optimization problems, always carefully define the variables and constraints, and make sure to check the critical points using the second derivative or a test for concavity to ensure they represent a minimum or maximum.