The side of a square is increasing at a rate of `3` cm/sec. Find the rate of increasing of its perimeter in cm/sec when the side of square is `5`cm.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand 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 rate of 3 cm/sec and the side is currently 5 cm.

### Step 2: Define Variables
Let:
- \( x \) = length of the side of the square (in cm)
- \( p \) = perimeter of the square (in cm)
- \( \frac{dx}{dt} \) = rate of change of the side length (in cm/sec)
- \( \frac{dp}{dt} \) = rate of change of the perimeter (in cm/sec)

Given:
- \( \frac{dx}{dt} = 3 \) cm/sec
- \( x = 5 \) cm

### Step 3: Write the Formula for Perimeter
The perimeter \( p \) of a square is given by the formula:
\[ p = 4x \]

### Step 4: Differentiate the Perimeter with Respect to Time
To find how the perimeter changes over time, we differentiate \( p \) with respect to \( t \):
\[ \frac{dp}{dt} = 4 \frac{dx}{dt} \]

### Step 5: Substitute the Known Values
Now, we substitute the known values into the differentiated equation:
\[ \frac{dp}{dt} = 4 \cdot \frac{dx}{dt} = 4 \cdot 3 \]

### Step 6: Calculate the Rate of Change of the Perimeter
Calculating the above expression:
\[ \frac{dp}{dt} = 12 \text{ cm/sec} \]

### Final Answer
The rate of increase of the perimeter when the side of the square is 5 cm is \( 12 \) cm/sec.

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