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?

$\frac{128}{3} \pi$

We know that,

Volume of cylinder = πr²h

Volume of sphere = \(\frac{4}{3}\)πR³

Curved surface area of cylinder = 2 πrh

We have,

The height of a cylinder = \(\frac{2}{3}\) of diameter.

Cylinder volume = volume of a sphere

Radius of the sphere = 4 cm.

Let radius of cylinder be 3a

Diameter of cylinder = 2 × 3a = 6a

Height of the cylinder = 6a × \(\frac{2}{3}\) = 4a

Radius of sphere R = 4 cm

According to the question

πr²h = \(\frac{4}{3}\)πR³

3a × 3a × 4a = \(\frac{4}{3}\) × 4 × 4 × 4

a³ = \(\frac{63}{27}\)

a = \(\frac{4}{3}\)

Curved surface area of cylinder = 24πa²= 24π(\(\frac{4}{3}\))² = $\frac{128}{3} \pi$