The length of a rectangle is increasing at the rate 3.5 cm/sec. and its breadth is decreasing at the rate of 3 cm/sec. Find the rate of change of the area of the rectangle when length is 12 cm and breadth is 8 cm.

Answer

To solve the problem, we need to find the rate of change of the area of a rectangle given the rates of change of its length and breadth. Let's break it down step by step.

### Step 1: Define the variables
Let:
- \( L \) = length of the rectangle
- \( B \) = breadth of the rectangle
- \( A \) = area of the rectangle

### Step 2: Write the formula for the area
The area \( A \) of a rectangle is given by the formula:
\[ A = L \times B \]

### Step 3: Differentiate the area with respect to time
To find the rate of change of the area with respect to time, we differentiate \( A \) using the product rule:
\[ \frac{dA}{dt} = L \frac{dB}{dt} + B \frac{dL}{dt} \]

### Step 4: Substitute the known values
From the problem, we know:
- \( \frac{dL}{dt} = 3.5 \) cm/sec (length is increasing)
- \( \frac{dB}{dt} = -3 \) cm/sec (breadth is decreasing)
- \( L = 12 \) cm
- \( B = 8 \) cm

Now we can substitute these values into the differentiated area formula:
\[ \frac{dA}{dt} = 12 \times (-3) + 8 \times 3.5 \]

### Step 5: Calculate the values
Now we calculate each term:
1. \( 12 \times (-3) = -36 \)
2. \( 8 \times 3.5 = 28 \)

Now, substitute these results back into the equation:
\[ \frac{dA}{dt} = -36 + 28 \]
\[ \frac{dA}{dt} = -8 \text{ cm}^2/\text{sec} \]

### Conclusion
The rate of change of the area of the rectangle when the length is 12 cm and the breadth is 8 cm is:
\[ \frac{dA}{dt} = -8 \text{ cm}^2/\text{sec} \]
This indicates that the area of the rectangle is decreasing at a rate of 8 cm²/sec.

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