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?
Answer & explanation
Correct answer: option 3
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 = $\frac{128}{3} \pi$