The side of a square is increasing at a rate of 4cm/sec. Find the rate of increase of its area when the side of square is 10 cm.

Answer

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

### Step 1: Define the variables
Let the side of the square be denoted as \( x \) cm. We know that the side of the square is increasing at a rate of \( \frac{dx}{dt} = 4 \) cm/sec.

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

### 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 \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt}(x^2) = 2x \frac{dx}{dt} \]

### Step 4: Substitute the known values
We need to find \( \frac{dA}{dt} \) when \( x = 10 \) cm and \( \frac{dx}{dt} = 4 \) cm/sec. Substituting these values into the differentiated equation:
\[ \frac{dA}{dt} = 2(10) \cdot (4) \]

### Step 5: Calculate the rate of increase of the area
Now, calculate:
\[ \frac{dA}{dt} = 20 \cdot 4 = 80 \text{ cm}^2/\text{sec} \]

### Conclusion
The rate of increase of the area of the square when the side is 10 cm is:
\[ \frac{dA}{dt} = 80 \text{ cm}^2/\text{sec} \]

---