The rate of change (in cm²/s) of the total surface area of a hemisphere with respect to radius r at r = 3 √1.331 cm is:

The correct option is 6.6π.

To find the rate of change of the total surface area of a hemisphere with respect to its radius, we use basic geometric formulas and differentiation.

Step 1: Formula for the Total Surface Area of a Hemisphere
The total surface area (A) of a solid hemisphere of radius r consists of its curved surface area (2πr²) and its flat circular base area (πr²). Therefore, the total surface area is:
$$A=3\pi r^2$$

Step 2: Differentiating Area with respect to Radius
We need to find the rate of change of the area A with respect to the radius r, which is given by the derivative dA/dr:
$$\frac{dA}{dr}=\frac{d}{dr}\left(3\pi r^2\right)$$
Using the power rule of differentiation:
$$\frac{dA}{dr}=6\pi r$$

Step 3: Calculating the Radius
The given radius r is the cube root of 1.331 cm:
$$r=\sqrt[3]{1.331}$$
Since 1.1³ = 1.331, we can simplify the value of r to:
$$r=1.1\text{cm}$$

Step 4: Evaluating the Rate of Change at the Given Radius
Substitute r = 1.1 cm into the derivative formula:
$$\frac{dA}{dr}=6\pi \left(1.1\right)=6.6\pi$$
Therefore, the rate of change of the total surface area of the hemisphere with respect to its radius is 6.6π.