What could one do to solve this optimization problem? **Question 8:**For any fixed number \( n \) representing the perimeter of a rectangle, what are the dimensions of the rectangle that maximizes the area of that rectangle?**Explanation:**To find the dimensions of the rectangle that maximizes the area given a fixed perimeter \( n \), we need to use the formula for the perimeter of a rectangle, \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width. Rearranging the formula gives \( l + w = \frac{n}{2} \).The area \( A \) of a rectangle is given by \( A = l \times w \).To maximize the area, substitute \( w = \frac{n}{2} - l \) into the area formula:\[ A = l \times \left(\frac{n}{2} - l\right) \]\[ A = \frac{n}{2}l - l^2 \]This quadratic equation can be simplified as:\[ A = -l^2 + \frac{n}{2}l \]To find the value of \( l \) that maximizes the area, we take the derivative of \( A \) with respect to \( l \) and set it to zero:\[ \frac{dA}{dl} = -2l + \frac{n}{2} = 0 \]Solving for \( l \), we get:\[ l = \frac{n}{4} \]Since \( l = w \) when the area is maximized, the rectangle becomes a square. Therefore, the dimensions of the rectangle that maximize the area are \( \frac{n}{4} \times \frac{n}{4} \).**Conclusion:**For a given perimeter \( n \), the dimensions of the rectangle that maximize the area are those of a square with sides each measuring \( \frac{n}{4} \).

Name: What could one do to solve this optimization problem? **Question 8:**F
Uploaded: 2024-03-20T07:07:26.000Z
Channel: Bartleby
Description: What could one do to solve this optimization problem? **Question 8:**For any fixed number \( n \) representing the perimeter of a rectangle, what are the dimensions of the rectangle that maximizes the area of that rectangle?**Explanation:**To find th