The side of a square is increasing at the rate of 0.5 cm/sec. Find the rate of increase of its area, when the side of square is 20 cm long.

Answer

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

### Step 1: Understand the given information
We know that:
- The side of the square (let's denote it as \( A \)) is increasing at a rate of \( \frac{dA}{dt} = 0.5 \) cm/sec.
- We need to find the rate of increase of the area of the square when the side length is \( A = 20 \) cm.

### Step 2: Write the formula for the area of the square
The area \( S \) of a square is given by the formula:
\[ S = A^2 \]

### Step 3: Differentiate the area with respect to time
To find the rate of change of the area with respect to time, we differentiate both sides of the area formula with respect to \( t \):
\[ \frac{dS}{dt} = \frac{d}{dt}(A^2) \]
Using the chain rule, we have:
\[ \frac{dS}{dt} = 2A \cdot \frac{dA}{dt} \]

### Step 4: Substitute the known values
Now we substitute the values we have:
- \( A = 20 \) cm
- \( \frac{dA}{dt} = 0.5 \) cm/sec

Substituting these values into the differentiated equation:
\[ \frac{dS}{dt} = 2 \cdot 20 \cdot 0.5 \]

### Step 5: Calculate the rate of increase of the area
Now we calculate:
\[ \frac{dS}{dt} = 2 \cdot 20 \cdot 0.5 = 20 \text{ cm}^2/\text{sec} \]

### Final Answer
The rate of increase of the area when the side of the square is 20 cm long is:
\[ \frac{dS}{dt} = 20 \text{ cm}^2/\text{sec} \]

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