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

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 is increasing at a rate of 3 cm/sec and the side length is 5 cm.

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

### Step 3: Write down the given information
From the problem, we know:
- \( \frac{dx}{dt} = 3 \) cm/sec (the rate at which the side is increasing)
- \( x = 5 \) cm (the length of the side at the moment we are interested in)

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

### Step 5: Differentiate the perimeter with respect to time
To find the rate of change of the perimeter with respect to time, we differentiate \( P \) with respect to \( t \):
\[ \frac{dP}{dt} = \frac{d}{dt}(4x) = 4 \frac{dx}{dt} \]

### Step 6: Substitute the known values
Now we substitute \( \frac{dx}{dt} = 3 \) cm/sec into the equation:
\[ \frac{dP}{dt} = 4 \times 3 = 12 \text{ cm/sec} \]

### Step 7: State the final answer
Thus, the rate of increase of the perimeter when the side of the square is 5 cm is:
\[ \frac{dP}{dt} = 12 \text{ cm/sec} \]