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

Answer

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

### Step 1: Understand the relationship between area and perimeter
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. The perimeter \(P\) of a rectangle is given by:
\[ P = 2(l + w) \]

### Step 2: Set the area equal to \(4 \, m^2\)
Since we know the area of the rectangle is \(4 \, m^2\), we can write:
\[ l \times w = 4 \]

### Step 3: Express one variable in terms of the other
We can express \(w\) in terms of \(l\):
\[ w = \frac{4}{l} \]

### Step 4: Substitute into the perimeter formula
Now, substitute \(w\) into the perimeter formula:
\[ P = 2\left(l + \frac{4}{l}\right) \]
This simplifies to:
\[ P = 2l + \frac{8}{l} \]

### Step 5: Find the minimum perimeter
To find the minimum perimeter, 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:
\[ 2 - \frac{8}{l^2} = 0 \]
Solving for \(l\):
\[ \frac{8}{l^2} = 2 \implies l^2 = 4 \implies l = 2 \, m \]

### Step 6: Calculate the corresponding width
Using \(l = 2 \, m\) in the area equation:
\[ w = \frac{4}{l} = \frac{4}{2} = 2 \, m \]

### Step 7: Calculate the minimum perimeter
Now, substituting \(l\) and \(w\) back into the perimeter formula:
\[ P = 2(l + w) = 2(2 + 2) = 2 \times 4 = 8 \, m \]

### Final Answer
The minimum possible perimeter for a rectangle whose area is \(4 \, m^2\) is:
\[ \boxed{8 \, m} \]

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