Let's tackle each part of the problem step-by-step.

(a) Writing the Function for the Total Area $A$ of the Page in Terms of $x$

1. The outer dimensions of the page are $x$ (width) and $y$ (height).

2. The inner dimensions of the printed area (taking into account the margins) are:

   • Width of the printed area: $x - 4$ (since each side has a 2-inch margin, totaling 4 inches).
   • Height of the printed area: $y - 2$ (since each top and bottom margin is 1 inch, totaling 2 inches).

3. The printed area is given as 30 square inches: $(x - 4)(y - 2) = 30$

4. Solving for $y$ in terms of $x$: $y - 2 = \frac{30}{x - 4} \Rightarrow y = \frac{30}{x - 4} + 2$

5. The total area $A$ of the page is: $A = x \cdot y$

6. Substituting $y$ from step 4 into this equation: $A = x \left( \frac{30}{x - 4} + 2 \right)$ Expanding this: $A = \frac{30x}{x - 4} + 2x$

So, the function for the total area $A$ of the page in terms of $x$ is: $A = \frac{30x}{x - 4} + 2x$

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(b) Determining the Domain of the Function

To determine the domain based on the physical constraints:

1. Since the inner width $x - 4$ must be positive, $x > 4$.
2. The printed area must fit within the outer page dimensions, so $x$ should be reasonably limited (e.g., if we assume the maximum dimension is 30 inches, then $x < 30$).

Thus, the domain of the function based on the constraints is: $x > 4$

(c) Graphing the Area Function and Finding Minimum Paper Use

Now, using a graphing utility to approximate the minimum value of $A$ within the domain $x > 4$:

1. Graph $A(x) = \frac{30x}{x - 4} + 2x$.
2. Locate the value of $x$ that minimizes $A(x)$ in the interval $x > 4$ and $x < 30$.
3. Calculate the corresponding $y$ value using $y = \frac{30}{x - 4} + 2$.

After graphing, you would find the values:

$x \approx \text{value to be determined by the graph}$ $y \approx \text{value to be determined by the graph}$

These approximate values will yield the dimensions of the page that use the least amount of paper.

Would you like help graphing the function or have any questions about these steps?

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Follow-up Questions

1. How does changing the area of print (e.g., to 40 square inches) affect the optimal dimensions?
2. What would happen to the function if the side margins were 3 inches instead of 2?
3. How does the total area $A$ behave as $x$ approaches 4?
4. Can we analyze the behavior of the function $A(x)$ using calculus?
5. How does the margin size impact the page dimensions for minimum paper use?

Tip: When graphing, pay attention to asymptotes and boundary conditions in the function, as these can reveal critical information about constraints and optimal values.