The perimeter of a rectangle is fixed at 24cm. If the length l of the rectangle is increasing at the rate of 1 cm per second, the value of l for which the area of rectangle start to decrease is

Options

A. 2cm
B. 6cm
C. 4cm
D. 8cm

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Understand the problem
We are given that the perimeter of a rectangle is fixed at 24 cm. The length \( l \) of the rectangle is increasing at a rate of 1 cm per second. We need to find the value of \( l \) for which the area of the rectangle starts to decrease.

### Step 2: Set up the equations
The perimeter \( P \) of a rectangle is given by:
\[ P = 2(l + b) = 24 \]
From this, we can express the breadth \( b \) in terms of the length \( l \):
\[ l + b = 12 \implies b = 12 - l \]

### Step 3: Write the area of the rectangle
The area \( A \) of the rectangle is given by:
\[ A = l \times b = l \times (12 - l) = 12l - l^2 \]

### Step 4: Differentiate the area with respect to time
To find when the area starts to decrease, we need to differentiate the area \( A \) with respect to time \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt}(12l - l^2) \]
Using the chain rule, we differentiate:
\[ \frac{dA}{dt} = 12 \frac{dl}{dt} - 2l \frac{dl}{dt} \]
Factoring out \( \frac{dl}{dt} \):
\[ \frac{dA}{dt} = \left(12 - 2l\right) \frac{dl}{dt} \]

### Step 5: Substitute the rate of change of length
We know that the length \( l \) is increasing at a rate of \( \frac{dl}{dt} = 1 \) cm/s. Substituting this into the equation gives:
\[ \frac{dA}{dt} = (12 - 2l) \cdot 1 = 12 - 2l \]

### Step 6: Determine when the area starts to decrease
The area starts to decrease when \( \frac{dA}{dt} < 0 \):
\[ 12 - 2l < 0 \]
Solving this inequality:
\[ 12 < 2l \implies l > 6 \]

### Step 7: Conclusion
The area of the rectangle starts to decrease when the length \( l \) is greater than 6 cm. Therefore, the value of \( l \) for which the area starts to decrease is:
\[ l > 6 \text{ cm} \]