The height of a cylinder is $$\frac{2}{3} rd$$ of its diameter. Its volume is equal to the volume of a sphere whose radius is 4 cm. What is the curved surface area (in cm$$^2$$) of the cylinder?

As per the question,

The height of the cylinder $$h=\dfrac{2D}{3}$$

Here D is the diameter of the cylinder, so $$R=\dfrac{D}{2}$$
As per the given condition in the question, Volume of the cylinder= Volume of the sphere,

$$\pi R^2 h=\dfrac{4\pi r^3}{3}-------(i)$$

Now, substituting the value of h in the equation (i)

$$\Rightarrow \pi \times (R)^2\times \dfrac{2\times 2R}{3}=\dfrac{4\pi r^3}{3}$$

$$\Rightarrow \dfrac{4R^3}{3}=\dfrac{4\times r^3}{3}$$

$$Rightarrow R^3=r^3$$

$$\Rightarrow R=4cm$$cm

So, curved surface area of the cylinder $$=2\pi R \times h=\dfrac{2 \pi 4\times 4\times 4}{3}=\dfrac{128\pi}{3}$$