The length x of a rectangle is decreasing at the rate of 5 cm/sec and the breadth y is increasing at the rate of 4 cm/sec. When x=8 cm and y=6 cm, find the rate of change of area of the rectangle.

Options

A. 2 cm/sec
B. `-2 ` cm/sec
C. `2 cm^(2)//sec`
D. `3 cm^2//sec`

Answer

To find the rate of change of the area of a rectangle when the length \( x \) is decreasing and the breadth \( y \) is increasing, we can follow these steps:

### Step 1: Understand the given rates of change
- The length \( x \) is decreasing at a rate of \( \frac{dx}{dt} = -5 \) cm/sec (negative because it's decreasing).
- The breadth \( y \) is increasing at a rate of \( \frac{dy}{dt} = 4 \) cm/sec.

### Step 2: Write down the formula for the area of the rectangle
The area \( A \) of a rectangle is given by the formula:
\[ A = x \cdot y \]

### Step 3: Differentiate the area with respect to time
To find the rate of change of the area, we differentiate \( A \) with respect to time \( t \):
\[ \frac{dA}{dt} = \frac{dx}{dt} \cdot y + x \cdot \frac{dy}{dt} \]

### Step 4: Substitute the known values
We know:
- \( x = 8 \) cm
- \( y = 6 \) cm
- \( \frac{dx}{dt} = -5 \) cm/sec
- \( \frac{dy}{dt} = 4 \) cm/sec

Now substitute these values into the differentiated equation:
\[ \frac{dA}{dt} = (-5) \cdot 6 + 8 \cdot 4 \]

### Step 5: Calculate the values
Now perform the calculations:
\[ \frac{dA}{dt} = -30 + 32 \]
\[ \frac{dA}{dt} = 2 \text{ cm}^2/\text{sec} \]

### Conclusion
The rate of change of the area of the rectangle is \( \frac{dA}{dt} = 2 \) cm²/sec, indicating that the area is increasing.