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, we need to find the rate of increase of the perimeter of a square when the side of the square is increasing at a certain rate. Let's break it down step by step.

### Step 1: Understand the relationship between the side length and perimeter
The perimeter \( P \) of a square is given by the formula:
\[ P = 4s \]
where \( s \) is the length of one side of the square.

### Step 2: Differentiate the perimeter with respect to time
To find the rate of change of the perimeter with respect to time, we differentiate both sides of the perimeter formula with respect to time \( t \):
\[ \frac{dP}{dt} = 4 \frac{ds}{dt} \]
where \( \frac{dP}{dt} \) is the rate of change of the perimeter and \( \frac{ds}{dt} \) is the rate of change of the side length.

### Step 3: Substitute the given rate of change of the side length
From the problem, we know that the side of the square is increasing at a rate of \( 0.2 \) cm/s. Therefore:
\[ \frac{ds}{dt} = 0.2 \, \text{cm/s} \]

### Step 4: Calculate the rate of change of the perimeter
Now, we can substitute \( \frac{ds}{dt} \) into the differentiated formula:
\[ \frac{dP}{dt} = 4 \times 0.2 \]
\[ \frac{dP}{dt} = 0.8 \, \text{cm/s} \]

### Conclusion
The rate of increase of the perimeter with respect to time is \( 0.8 \, \text{cm/s} \).

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