Correct Answer: $\frac{128}{3} \pi$

Solution : Given,
The height of a cylinder is $\frac23$rd of its diameter.
Cylinder volume is equal to the volume of a sphere whose radius is 4 cm.
We know,
Volume of cylinder = $\pi r^2h$
Volume of sphere = $\frac43\pi R^3$
Curved surface area of cylinder = $2\pi rh$
Where, $r$ = radius of the cylinder and $R$ = Radius of the sphere
Let the radius of the cylinder be $3x$
⇒ Diameter of cylinder = $2 × 3x$ = $6x$
Height of the cylinder = $6x × \frac23$ = $4x$
The radius of sphere $R$ = 4 cm
According to the question,
$\pi r^2h =\frac43\pi R^3$
⇒ $3x × 3x × 4x = \frac43 × 4 × 4 × 4$
⇒ $x^3 = \frac{4 × 4 × 4}{3 × 3 × 3}$
⇒ $x = \sqrt[3]{\frac{4 × 4 × 4}{3 × 3 × 3}}$
⇒ $x=\frac43$
Curved surface area of cylinder = $2 \pi rh$
= $2π × 3x × 4x$
= $2π × 3 × \frac43 × 4 × \frac43$
= $\frac{128}{3} \pi$
Hence, the correct answer is $\frac{128}{3}\pi$.