We consider the energy-minimizing configuration of the Sadowsky-type functional for narrow rectifying strips. We show that the functional is proportional to the $p$-Willmore functional using classical analysis techniques and the geometry of developable surfaces. We introduce hyperelastic strips (or p-elastic strips) as rectifying strips whose base curves are the critical points of the Sadowsky-type functional and find the Euler-Lagrange equations for hyperelastic strips using a variational approach. We show a naturally expected relationship between the planar stationary points of the Sadowsky-type functional and the hyperelastic curves. We derive two conservation vector fields, the internal force and torque, using Euclidean motions and obtain the first and second conservation laws for hyperelastic strips.