The rate of change of area of a circle with respect to its circumference when radius is 4cm, is

The correct option is 4 cm²/cm.

To find the rate of change of the area of a circle with respect to its circumference, we can define both quantities as functions of the radius, r, and then use the chain rule of differentiation.

Let A denote the area of the circle. The formula for the area of a circle in terms of its radius r is:

$$A=\pi r^2$$

Differentiating the area A with respect to the radius r, we get:

$$\frac{dA}{dr}=\frac{d}{dr}\left(\pi r^2\right)=2\pi r$$

Next, let C denote the circumference of the circle. The formula for the circumference in terms of the radius r is:

$$C=2\pi r$$

Differentiating the circumference C with respect to the radius r, we obtain:

$$\frac{dC}{dr}=\frac{d}{dr}\left(2\pi r\right)=2\pi$$

Now, to find the rate of change of area A with respect to circumference C, we compute the derivative of A with respect to C:

$$\frac{dA}{dC}=\frac{dA/dr}{dC/dr}$$

Substituting the values of the derivatives we calculated earlier:

$$\frac{dA}{dC}=\frac{2\pi r}{2\pi }=r$$

We are given that the radius of the circle is 4 cm (r = 4 cm). Substituting this value into our equation:

$$\frac{dA}{dC}=4\text{ cm}$$

Thus, the rate of change of the area of the circle with respect to its circumference is 4 cm²/cm.