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We are tasked with determining the dimensions of a poster that maximizes the printing area, given specific constraints. Let's break down the problem step by step:

### Problem Breakdown:
- The **maximum perimeter** of the poster is 42 inches.
- The **printable area** cannot extend within:
  - 1 inch from the top and bottom.
  - 1.5 inches from the sides.
  
Thus, the printable area is smaller than the total area of the poster by 2 inches vertically (1 inch from both top and bottom) and by 3 inches horizontally (1.5 inches from both sides).

### Step 1: Defining Variables
Let:
- \( L \) be the **length** of the poster.
- \( W \) be the **width** of the poster.

The perimeter condition gives us the equation:
\[
2L + 2W = 42
\]
or equivalently:
\[
L + W = 21
\]

The **dimensions of the printable area** are:
- **Length** of the printable area: \( L - 2 \) (since we subtract 2 inches: 1 inch from both the top and bottom).
- **Width** of the printable area: \( W - 3 \) (since we subtract 3 inches: 1.5 inches from both sides).

### Step 2: Area of Printable Region
The area \( A \) of the printable region is:
\[
A = (L - 2)(W - 3)
\]

### Step 3: Maximizing the Area
We now express \( A \) as a function of one variable using the perimeter equation \( L + W = 21 \). From this equation, we can express \( L \) in terms of \( W \):
\[
L = 21 - W
\]
Substitute this into the area formula:
\[
A(W) = (21 - W - 2)(W - 3) = (19 - W)(W - 3)
\]
Expand this expression:
\[
A(W) = 19W - 57 - W^2 + 3W = -W^2 + 22W - 57
\]
Thus, the area function is:
\[
A(W) = -W^2 + 22W - 57
\]

### Step 4: Finding the Maximum
To maximize the area, we can take the derivative of \( A(W) \) with respect to \( W \), set it equal to zero, and solve for \( W \):
\[
\frac{dA}{dW} = -2W + 22
\]
Setting this equal to zero:
\[
-2W + 22 = 0
\]
Solving for \( W \):
\[
W = 11
\]

### Step 5: Finding \( L \)
Substitute \( W = 11 \) back into the perimeter equation \( L + W = 21 \):
\[
L + 11 = 21 \quad \Rightarrow \quad L = 10
\]

### Final Answer:
The overall dimensions of the poster that maximize the printable area are:
\[
\boxed{10 \text{ inches} \text{ by } 11 \text{ inches}}
\]

Would you like a more detailed explanation of any step, or have further questions?

Here are 5 related questions to expand on this:
1. How does changing the perimeter affect the optimal dimensions of the poster?
2. What would the dimensions be if the margin requirements were different (e.g., larger margins)?
3. How can we verify that these dimensions indeed give the maximum area?
4. How would the solution change if the poster were square-shaped?
5. Can this approach be generalized for any poster with different margin and perimeter constraints?

**Tip:** When dealing with optimization problems like this one, always remember to express the target function (here, the area) in terms of a single variable for easier differentiation.

A poster can have a maximum perimeter of 42 inches. The area in which an image can be printed can be no closer than 1 inch from the top and bottom of the poster and no closer than 1.5 inches from the sides. What should the overall dimensions of the poster be in order to maximize the printing area?

Solve a problem involving maximizing the printing area of a poster with a maximum perimeter of 42 inches and specific margin constraints. Learn how to use quadratic equations and optimization techniques to find the solution.

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