The adjacent sides of a rectangle with given perimeter as 100 cm and enclosing maximum area are

Options

A. 10 cm and 40 cm
B. 20 cm and 30 cm
C. 25 cm and 25 cm
D. 15 cm and 35 cm

Explanation

To solve the problem of finding the adjacent sides of a rectangle with a given perimeter of 100 cm that encloses the maximum area, we can follow these steps:

### Step 1: Understand the perimeter of the rectangle
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2A + 2B \]
where \( A \) and \( B \) are the lengths of the adjacent sides of the rectangle.

### Step 2: Set up the equation for the perimeter
Given that the perimeter is 100 cm, we can write:
\[ 2A + 2B = 100 \]
Dividing the entire equation by 2 gives:
\[ A + B = 50 \]
This is our first relation.

### Step 3: Express one variable in terms of the other
From the equation \( A + B = 50 \), we can express \( A \) in terms of \( B \):
\[ A = 50 - B \]

### Step 4: Write the area of the rectangle
The area \( A \) of the rectangle is given by:
\[ \text{Area} = A \times B \]
Substituting \( A = 50 - B \) into the area formula gives:
\[ \text{Area} = (50 - B) \times B = 50B - B^2 \]

### Step 5: Differentiate the area function
To find the maximum area, we need to differentiate the area function with respect to \( B \):
\[ \frac{d(\text{Area})}{dB} = 50 - 2B \]

### Step 6: Set the derivative equal to zero
To find the critical points, set the derivative equal to zero:
\[ 50 - 2B = 0 \]
Solving for \( B \) gives:
\[ 2B = 50 \quad \Rightarrow \quad B = 25 \]

### Step 7: Find the corresponding value of A
Now, substitute \( B = 25 \) back into the equation \( A = 50 - B \):
\[ A = 50 - 25 = 25 \]

### Step 8: Verify that this gives a maximum area
To confirm that this critical point gives a maximum area, we can check the second derivative:
\[ \frac{d^2(\text{Area})}{dB^2} = -2 \]
Since the second derivative is negative, this indicates that we have a maximum area at \( B = 25 \).

### Conclusion
Thus, the adjacent sides of the rectangle that enclose the maximum area with a perimeter of 100 cm are:
\[ A = 25 \, \text{cm}, \quad B = 25 \, \text{cm} \]