The rate of change of area of a square is 40 cm2/s. What will be the rate of change of side if the side is 10 cm

The correct option is 2 cm/s.

Let us break down the solution step-by-step using the concepts of derivatives and rates of change.

Step 1: Define the variables and formula
Let the side length of the square be represented by _s and its area by _A.
The formula for the area of a square in terms of its side length is:

$$A=s^2$$

Step 2: Differentiate with respect to time
To find how the area and the side length change over time, we differentiate both sides of the equation with respect to time ($t$) using the chain rule:

$$\frac{dA}{dt}=\frac{d}{dt}\left(s^2\right)$$
Applying the chain rule gives:

$$\frac{dA}{dt}=2s\cdot \frac{ds}{dt}$$
Here, $\frac{dA}{dt}$ represents the rate of change of the area, and $\frac{ds}{dt}$ represents the rate of change of the side length.

Step 3: Substitute the given values
We are given:
• The rate of change of the area, $\frac{dA}{dt}=40{\text{cm}}^2\text{/s}$
• The side length of the square at the instant of interest, $s=10\text{cm}$

Plugging these values into our differentiated equation:

$$40=2\left(10\right)\cdot \frac{ds}{dt}$$
$$40=20\cdot \frac{ds}{dt}$$

Step 4: Solve for the rate of change of the side
Rearranging the equation to find $\frac{ds}{dt}$:

$$\frac{ds}{dt}=\frac{40}{20}$$
$$\frac{ds}{dt}=2\text{cm/s}$$
Thus, the side length of the square is increasing at a rate of 2 cm/s.