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The problem is asking us to find the dimensions of a rectangle with the largest area, given that its perimeter is 188. Let's go step by step.

### Given:
- Perimeter \( P = 188 \)
- We need to maximize the area of the rectangle.

### Step 1: Relationship between dimensions and perimeter
Let the dimensions of the rectangle be \( l \) (length) and \( w \) (width). The perimeter of a rectangle is given by:
\[
P = 2l + 2w
\]
Substitute \( P = 188 \):
\[
188 = 2l + 2w
\]
Dividing both sides by 2:
\[
94 = l + w
\]
Thus, we have:
\[
l = 94 - w
\]

### Step 2: Area of the rectangle
The area \( A \) of the rectangle is given by:
\[
A = l \times w
\]
Substitute \( l = 94 - w \) into the area formula:
\[
A = (94 - w) \times w
\]
Expanding:
\[
A = 94w - w^2
\]

### Step 3: Maximizing the area
The area function \( A = 94w - w^2 \) is a quadratic equation. To find the maximum value of \( A \), we find the vertex of the parabola, which occurs at:
\[
w = \frac{-b}{2a}
\]
In the equation \( A = -w^2 + 94w \), the coefficients are \( a = -1 \) and \( b = 94 \). So:
\[
w = \frac{-94}{2 \times -1} = \frac{94}{2} = 47
\]

### Step 4: Finding the dimensions
When \( w = 47 \), substitute this back into \( l = 94 - w \) to find the length:
\[
l = 94 - 47 = 47
\]

Thus, the rectangle with the largest area is a **square** with dimensions \( 47 \times 47 \).

### Step 5: Writing the answer as fractions
Since the dimensions are already whole numbers, the answer in fractional form (if necessary) would be:
\[
\frac{47}{1} \times \frac{47}{1}
\]

### Final Answer:
The dimensions of the rectangle with the largest area are \( \frac{47}{1} \times \frac{47}{1} \) or simply \( 47 \times 47 \).

Would you like more details on this? Here are some related questions to explore:

1. How do you determine the maximum or minimum of a quadratic function?
2. What is the significance of the fact that the largest-area rectangle is a square?
3. How can we use calculus to verify that \( w = 47 \) gives a maximum area?
4. How does the perimeter affect the shape of a rectangle with maximum area?
5. What are some practical applications of this type of optimization problem?

**Tip:** In optimization problems, if you're maximizing or minimizing a quantity, always check if the function you're working with is quadratic, as it will simplify finding the solution!

Among all rectangles that have a perimeter of 188, find the dimensions of the one whose area is largest. Write your answers as fractions reduced to lowest terms.

Find the dimensions of a rectangle with a fixed perimeter of 188 that maximizes its area. This involves solving a quadratic equation and exploring the relationship between perimeter and area. Suitable for high school students in Grades 9-10.

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