What is the minimum possible possible perimeter for a rectangle whose area is `4 m^(2)` ?

Answer

To find the minimum possible perimeter for a rectangle with a given area of \(4 \, m^2\), we can follow these steps:

### Step 1: Understand the relationship between area and dimensions
The area \(A\) of a rectangle is given by the formula:
\[ A = l \times w \]
where \(l\) is the length and \(w\) is the width of the rectangle.

### Step 2: Set up the equation for the area
Given that the area is \(4 \, m^2\), we can write:
\[ l \times w = 4 \]

### Step 3: Express the perimeter in terms of length and width
The perimeter \(P\) of a rectangle is given by the formula:
\[ P = 2(l + w) \]

### Step 4: Substitute \(w\) in terms of \(l\)
From the area equation, we can express \(w\) as:
\[ w = \frac{4}{l} \]
Now substitute this into the perimeter formula:
\[ P = 2\left(l + \frac{4}{l}\right) \]

### Step 5: Simplify the perimeter equation
Now, we can simplify the perimeter equation:
\[ P = 2l + \frac{8}{l} \]

### Step 6: Find the minimum perimeter using calculus
To find the minimum value of \(P\), we can take the derivative of \(P\) with respect to \(l\) and set it to zero:
\[ \frac{dP}{dl} = 2 - \frac{8}{l^2} \]
Setting the derivative equal to zero gives:
\[ 2 - \frac{8}{l^2} = 0 \]
\[ \frac{8}{l^2} = 2 \]
\[ l^2 = 4 \quad \Rightarrow \quad l = 2 \]

### Step 7: Calculate the corresponding width
Now, substitute \(l = 2\) back into the area equation to find \(w\):
\[ w = \frac{4}{2} = 2 \]

### Step 8: Calculate the minimum perimeter
Now that we have both \(l\) and \(w\) as \(2\), we can calculate the perimeter:
\[ P = 2(l + w) = 2(2 + 2) = 2 \times 4 = 8 \, m \]

### Conclusion
The minimum possible perimeter for a rectangle with an area of \(4 \, m^2\) is:
\[ \boxed{8 \, m} \]