A rectangle is constructed such that the sum of its length
and width is 54 . Let the length of the rectangle be .
Form an expression for , the area of the rectangle, in terms of .
What length of rectangle will result in the maximum possible area?
Length
What is the width of the rectangle with maximum area?
Width = Enter your next step here

Ask by Collins Martin. in the United States

Jan 21,2025

Question

A rectangle is constructed such that the sum of its length
and width is 54 . Let the length of the rectangle be .
Form an expression for , the area of the rectangle, in terms of .
What length of rectangle will result in the maximum possible area?
Length
What is the width of the rectangle with maximum area?
Width = Enter your next step here

Ask by Collins Martin. in the United States

Jan 21,2025

UpStudy AI · Accepted Answer

To determine the width of the rectangle that results in the maximum possible area, let's follow these steps:

1. **Express the Width in Terms of Length:**
   
   Given that the sum of the length ($$ x $$) and width is 54, we can express the width ($$ w $$) as:
   
   $$
   w = 54 - x
   $$
   
2. **Form the Area Function:**
   
   The area ($$ A $$) of the rectangle is the product of its length and width:
   
   $$
   A = x \cdot w = x \cdot (54 - x) = 54x - x^2
   $$
   
3. **Find the Length that Maximizes the Area:**
   
   The area function $$ A = 54x - x^2 $$ is a quadratic equation that opens downward, meaning its maximum value occurs at the vertex. The $$ x $$-coordinate of the vertex is found using:
   
   $$
   x = -\frac{b}{2a}
   $$
   
   Here, $$ a = -1 $$ and $$ b = 54 $$:
   
   $$
   x = -\frac{54}{2(-1)} = 27
   $$
   
   So, the length that maximizes the area is **27 units**.
   
4. **Determine the Width Corresponding to This Length:**
   
   Using the expression for width:
   
   $$
   w = 54 - x = 54 - 27 = 27
   $$
   
   Therefore, the width of the rectangle with the maximum area is **27 units**.

**Answer:**  
The width of the rectangle with maximum area is **27 units**.
The width of the rectangle with maximum area is 27 units.

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