Maximum area of a rectangle whose perimeter is given as 24 metres is equal to

Options

A. `36m^2`
B. `49m^2`
C. `64m^2`
D. `81m^2`

Answer

To find the maximum area of a rectangle with a given perimeter of 24 meters, we can follow these steps:

### Step 1: Understand the perimeter formula
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(L + B) \]
where \( L \) is the length and \( B \) is the breadth of the rectangle. Given that the perimeter is 24 meters, we can set up the equation:
\[ 2(L + B) = 24 \]

### Step 2: Simplify the equation
Dividing both sides of the equation by 2 gives:
\[ L + B = 12 \]

### Step 3: Express the area in terms of one variable
The area \( A \) of the rectangle is given by:
\[ A = L \times B \]
From the equation \( L + B = 12 \), we can express \( B \) in terms of \( L \):
\[ B = 12 - L \]
Now substitute this into the area formula:
\[ A = L(12 - L) = 12L - L^2 \]

### Step 4: Find the derivative of the area
To find the maximum area, we need to take the derivative of the area function \( A \) with respect to \( L \):
\[ \frac{dA}{dL} = 12 - 2L \]

### Step 5: Set the derivative to zero
To find the critical points, set the derivative equal to zero:
\[ 12 - 2L = 0 \]
Solving for \( L \):
\[ 2L = 12 \implies L = 6 \]

### Step 6: Find the corresponding breadth
Using the value of \( L \) to find \( B \):
\[ B = 12 - L = 12 - 6 = 6 \]

### Step 7: Calculate the maximum area
Now, substitute \( L \) and \( B \) back into the area formula:
\[ A = L \times B = 6 \times 6 = 36 \text{ square meters} \]

### Conclusion
The maximum area of the rectangle with a perimeter of 24 meters is:
\[ \boxed{36 \text{ square meters}} \]