Cost of truss is inversely proportional to the spacing of truss,

∴ t = k₁/s

Cost of purlins is directly proportional to the square of spacing of trusses

∴ P = k₂/s²

Cost of roof covering is directly proportional to the spacing of trusses

              ∴r = k₃s 

∴ Total cost = t + p + r

C = k₁/s + k₂s² + k₃s

For minimum cost, dc/ds = 0

\-k₁/s² + 2k₂s + k₃ = 0 

⇒ k₁/s + 2k₂s² + k₃s = 0 

⇒ −t+2p+ r = 0 

\ t = 2p + r