A rectangle with perimeter 32 cm has greatest area if its length is

Options

A. 12
B. 10
C. 8
D. 14

Answer

To find the length of a rectangle with a perimeter of 32 cm that has the greatest 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:
\[ P = 2(x + y) \]
where \( x \) is the length and \( y \) is the breadth of the rectangle. Given that the perimeter is 32 cm, we can set up the equation:
\[ 2(x + y) = 32 \]

### Step 2: Simplify the perimeter equation
Dividing both sides of the equation by 2 gives:
\[ x + y = 16 \]
From this, we can express \( y \) in terms of \( x \):
\[ y = 16 - x \]

### Step 3: Write the area of the rectangle
The area \( A \) of the rectangle can be expressed as:
\[ A = x \cdot y \]
Substituting \( y \) from the previous step:
\[ A = x(16 - x) = 16x - x^2 \]

### Step 4: Differentiate the area function
To find the maximum area, we need to differentiate the area function with respect to \( x \):
\[ \frac{dA}{dx} = 16 - 2x \]

### Step 5: Set the derivative equal to zero
To find the critical points, set the derivative equal to zero:
\[ 16 - 2x = 0 \]
Solving for \( x \) gives:
\[ 2x = 16 \implies x = 8 \]

### Step 6: Verify if it is a maximum
To confirm that this is a maximum, we can take the second derivative:
\[ \frac{d^2A}{dx^2} = -2 \]
Since the second derivative is negative, this indicates that the area function has a maximum at \( x = 8 \).

### Step 7: Find the value of \( y \)
Now, substituting \( x = 8 \) back into the equation for \( y \):
\[ y = 16 - x = 16 - 8 = 8 \]

### Conclusion
Thus, the length of the rectangle that gives the greatest area is:
\[ \text{Length} = 8 \text{ cm} \]