The length of a rectangle is decreasing at a rate of 3 cm/sec and breadth is increasing at a rate of 4 cm/sec. Find the rate of change of its (a) perimeter (b) area, when the length and breadth of rectangle are 7 cm and 8 cm respectively.

Answer

To solve the problem, we will follow these steps:

### Given:
- Length of rectangle (x) = 7 cm
- Breadth of rectangle (y) = 8 cm
- Rate of change of length (dx/dt) = -3 cm/sec (decreasing)
- Rate of change of breadth (dy/dt) = 4 cm/sec (increasing)

### (a) Finding the rate of change of the perimeter:

1. **Formula for the perimeter (P)** of a rectangle:
\[ P = 2(x + y) \]

2. **Differentiate the perimeter with respect to time (t)**:
\[ \frac{dP}{dt} = 2\left(\frac{dx}{dt} + \frac{dy}{dt}\right) \]

3. **Substitute the values of dx/dt and dy/dt**:
\[ \frac{dP}{dt} = 2\left(-3 + 4\right) \]

4. **Calculate**:
\[ \frac{dP}{dt} = 2(1) = 2 \text{ cm/sec} \]

### (b) Finding the rate of change of the area:

1. **Formula for the area (A)** of a rectangle:
\[ A = x \cdot y \]

2. **Differentiate the area with respect to time (t)** using the product rule:
\[ \frac{dA}{dt} = x \frac{dy}{dt} + y \frac{dx}{dt} \]

3. **Substitute the known values**:
- \(x = 7\) cm
- \(y = 8\) cm
- \(\frac{dy}{dt} = 4\) cm/sec
- \(\frac{dx}{dt} = -3\) cm/sec

\[ \frac{dA}{dt} = 7(4) + 8(-3) \]

4. **Calculate**:
\[ \frac{dA}{dt} = 28 - 24 = 4 \text{ cm}^2/\text{sec} \]

### Final Answers:
- Rate of change of the perimeter = **2 cm/sec**
- Rate of change of the area = **4 cm²/sec**