The side of a square is increasing at the rate of 0.2 cm/s. The rate of increase of perimeter w.r.t time is :

Options

A. 0.2 cm/s
B. 0.4 cm/s
C. 0.6 cm/s
D. 0.8 cm/s

Explanation

To solve the problem of finding the rate of increase of the perimeter of a square when the side is increasing at a rate of 0.2 cm/s, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the variables**:
- Let the length of the side of the square be denoted as \( x \).
- The rate of increase of the side with respect to time is given as \( \frac{dx}{dt} = 0.2 \) cm/s.

2. **Write the formula for the perimeter of a square**:
- The perimeter \( P \) of a square is given by the formula:
\[ P = 4x \]

3. **Differentiate the perimeter with respect to time**:
- To find the rate of change of the perimeter with respect to time, we differentiate the perimeter equation with respect to \( t \):
\[ \frac{dP}{dt} = \frac{d}{dt}(4x) = 4 \frac{dx}{dt} \]

4. **Substitute the known rate of change of the side**:
- Now, substitute \( \frac{dx}{dt} = 0.2 \) cm/s into the differentiated equation:
\[ \frac{dP}{dt} = 4 \times 0.2 \]

5. **Calculate the rate of increase of the perimeter**:
- Perform the multiplication:
\[ \frac{dP}{dt} = 0.8 \text{ cm/s} \]

### Final Answer:
The rate of increase of the perimeter with respect to time is \( 0.8 \) cm/s.

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