#### To determine

**To find:** The rate is the area of square increasing when the area of the square is 16 cm2.

#### Answer

The area of square increasing at the rate of dAdt=48 cm2/s.

#### Explanation

**Given:**

Each side of the square is increasing at a rate of 6 cm\s.

**Derivative rules:**

*Chain rule*: dydx=dydu⋅dudx

**Calculation:**

Let *A* be an area of the square with length of the side *x* and time variable is *t*.

Here, the side of the square is increasing as time is increasing. That is, the side of the square is depends on the time variable.

Thus, the side of the square is a function of time *t*.

The area of the square is depends on its length of side. So the area of the square is also function of the time since the side of the square is a function of time *t*.

Note that, the rates of change is a derivatives.

Each side of the square is increasing at a rate of 6 cm\s. That is, dxdt=6 m\s.

Obtain the derivatives dAdt when area of the square 16 cm2.

The area of square is x2 this implies that x2=16.

That is, x=4.

One side of the square *x* is 4 cm when area of the square is 16 cm2.

The area of the square A=x2.

Differentiate with respect to time variable *t*.

dAdt=ddt(x2)

Apply the chain rule and simplify the terms,

dAdt=ddx(x2)dxdt=2xdxdt

Substitute the value x=4 cm and dxdt=6 cm/s in dAdt=2xdxdt,

dAdt=2(4 cm)(6 cm/s)=48 cm2/s

Therefore, the area of square increasing at the rate of dAdt=48 cm2/s.