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 will use the relationship between the side of the square and its area, and apply the concept of derivatives.

### Step-by-Step Solution:

1. **Identify the Variables:**
Let \( x \) be the length of the side of the square (in cm).
The area \( A \) of the square is given by the formula:
\[ A = x^2 \]

2. **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 time \( t \):
\[ \frac{dA}{dt} = \frac{d}{dt}(x^2) = 2x \frac{dx}{dt} \]
Here, \( \frac{dA}{dt} \) is the rate of change of the area, and \( \frac{dx}{dt} \) is the rate of change of the side length.

3. **Substitute the Known Values:**
We know from the problem that:
- \( \frac{dx}{dt} = 4 \) cm/sec (the rate at which the side is increasing)
- We need to find \( \frac{dA}{dt} \) when \( x = 10 \) cm.

Now substitute these values into the differentiated equation:
\[ \frac{dA}{dt} = 2(10) \cdot 4 \]

4. **Calculate the Rate of Increase of the Area:**
Now, perform the multiplication:
\[ \frac{dA}{dt} = 20 \cdot 4 = 80 \text{ cm}^2/\text{sec} \]

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