We are tasked with minimizing the amount of paper used for a rectangular page, given the following conditions:

• The printed area must be 162 square inches.
• The top and bottom margins are each 2 inches wide.
• The side margins are each 1 inch wide.

Step 1: Define the problem in terms of $x$ and $y$

Let:

• $x$ = total width of the page,
• $y$ = total height of the page.

The printed area (i.e., the region without margins) will have dimensions:

• Width of printed area = $x - 2$ (since 1 inch margins on each side reduce the total width by 2 inches),
• Height of printed area = $y - 4$ (since 2 inch margins on top and bottom reduce the height by 4 inches).

We are told the printed area is 162 square inches, so: $(x - 2)(y - 4) = 162.$ This equation represents the area constraint.

Step 2: Express the area of the page

The area $A$ of the page (total area including margins) is: $A = x \cdot y.$ Our goal is to minimize this total area.

Step 3: Solve the area constraint equation for $y$

From the constraint: $(x - 2)(y - 4) = 162,$ solve for $y$: $y - 4 = \frac{162}{x - 2},$ $y = \frac{162}{x - 2} + 4.$

Step 4: Substitute into the total area formula

Now substitute $y$ into the total area formula $A = x \cdot y$: $A = x \left( \frac{162}{x - 2} + 4 \right).$ Simplify this expression: $A = x \cdot \frac{162}{x - 2} + 4x,$ $A = \frac{162x}{x - 2} + 4x.$ This is the function we want to minimize.

Step 5: Minimize the area function

To minimize $A(x)$, take the derivative of $A$ with respect to $x$, and set it equal to zero: $\frac{dA}{dx} = \frac{d}{dx} \left( \frac{162x}{x - 2} + 4x \right).$ Using the quotient rule for the first term: $\frac{d}{dx} \left( \frac{162x}{x - 2} \right) = \frac{(x - 2)(162) - 162x(1)}{(x - 2)^2} = \frac{162x - 324 - 162x}{(x - 2)^2} = \frac{-324}{(x - 2)^2}.$ Thus, the derivative is: $\frac{dA}{dx} = \frac{-324}{(x - 2)^2} + 4.$ Set this equal to zero: $\frac{-324}{(x - 2)^2} + 4 = 0,$ $\frac{-324}{(x - 2)^2} = -4,$ $(x - 2)^2 = \frac{324}{4} = 81,$ $x - 2 = 9,$ $x = 11.$

Step 6: Find $y$

Now that we have $x = 11$, substitute this into the equation for $y$: $y = \frac{162}{x - 2} + 4 = \frac{162}{11 - 2} + 4 = \frac{162}{9} + 4 = 18 + 4 = 22.$

Step 7: Conclusion

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

• Width $x = 11$ inches,
• Height $y = 22$ inches.

Would you like more details or have any questions?

Relative questions:

1. How do you apply the method of Lagrange multipliers for optimization problems?
2. Can you explain how the quotient rule works for derivatives?
3. What is the significance of taking the derivative in optimization problems?
4. How do margin constraints affect optimization in design problems?
5. What other real-life problems can be solved using area minimization?

Tip:

Always check the second derivative to ensure you’ve found a minimum, not a maximum, in optimization problems!