Find the length of the loop for the given curve:

x = 12t - 4t^ (3), y = 12t^ (2)

when x = 0, the vaues of t are 0 and ± 1/√3.

The arc length, S is determined by

∫ √[ (dx/dt) ^2 + (dy/dt) ^2] dt

= 2 ∫ √[ (12 - 12t^2) ^2 + (24t) ^2] dt from t = 0 to t = 1/√3

= 2 ∫ 12 √[ (1 - t^2) ^2 + (2t) ^2] dt from t = 0 to t = 1/√3

= 24 ∫ √ (1 + 2t^2 + t^4) dt from t = 0 to t = 1/√3

= 24 ∫ √ (1 + t^2) ^2 dt from t = 0 to t = 1/√3

= 24 ∫ (1 + t^2) dt from t = 0 to t = 1/√3

= 24 (t + t^3/3) from t = 0 to t = 1/√3

= 24 * (1/√3) * (1 + 1/9)

= 24 * (√3/3) * (10/9)

S = (80/9) √3