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 find the adjacent sides of a rectangle that has a given perimeter of 100 cm and encloses the maximum area, we can follow these steps:

### Step 1: Define the Variables
Let the lengths of the adjacent sides of the rectangle be \( x \) and \( y \).

### Step 2: Write the Perimeter Equation
The perimeter \( P \) of a rectangle is given by:
\[ P = 2(x + y) \]
Given that the perimeter is 100 cm, we can set up the equation:
\[ 2(x + y) = 100 \]
Dividing both sides by 2, we get:
\[ x + y = 50 \]

### Step 3: Express Area in Terms of One Variable
The area \( A \) of the rectangle is given by:
\[ A = x \cdot y \]
From the equation \( x + y = 50 \), we can express \( y \) in terms of \( x \):
\[ y = 50 - x \]
Substituting this into the area formula gives:
\[ A = x(50 - x) = 50x - x^2 \]

### Step 4: Differentiate the Area Function
To find the maximum area, we need to differentiate the area function with respect to \( x \):
\[ \frac{dA}{dx} = 50 - 2x \]

### Step 5: Set the Derivative to Zero
To find the critical points, we set the derivative equal to zero:
\[ 50 - 2x = 0 \]
Solving for \( x \):
\[ 2x = 50 \implies x = 25 \]

### Step 6: Find the Corresponding Value of \( y \)
Now, using the value of \( x \) to find \( y \):
\[ y = 50 - x = 50 - 25 = 25 \]

### Step 7: Conclusion
Thus, the adjacent sides of the rectangle that encloses the maximum area are:
\[ x = 25 \text{ cm} \quad \text{and} \quad y = 25 \text{ cm} \]
This means that the rectangle is actually a square.

### Final Answer
The adjacent sides of the rectangle are both 25 cm.

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