Let the Radius of the solid cylinder be ‘r’ m and its height be ‘h’ m.

Given,

Sum of radius and height of solid cylinder = 37 m

r + h = 37 m

r = 37 – h

Total surface area of solid cylinder = 1628 m²

Total surface area of solid cylinder is given by 2πr (h + r)

∴ 2πr (h + r) =1628 m²

Substituting the value of r + h in the above equation

⇒ 2πr × 37 = 1628 m²

⇒ r = 1628 × 7/22 × 1/2 × 1/37 m

⇒ r = 7 m

Since, r + h = 37 m

h = 37 – r m

h = 37 – 7 m = 30 m

Volume of solid cylinder = πr²h

= 22/7 × 7² × 30 m²

= 4620 m²