Given: The ratio between the radius of the base and the height of the cylinder is 2: 3.

Volume of the Cylinder is 1617cm³

Let 2x and 3x be radius and height of the Cylinder respectively.

Volume of the Cylinder is given as: πr²h

∴ πr²h = 1617

⇒ π × (2x)² × (3x) = 1617

⇒ 12π × x³ = 1617

⇒ x³ = = =

⇒ x = ∛ () =

∴ r = 2 × = 7cm

and, h = 3 × = 10.5 cm

Total surface area of a cylinder is: 2πrh(r + h)

Let S be the TSA of a cylinder

∴ S = 2πrh(r + h)

⇒ S = 2 × π × (7) × (7 + 10.5) = 2 × π × (7) × 17.5 = 770

∴ S = 770cm²

That is, Total surface area of the cylinder is 770 cm²