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A crate, initially traveling horizontally with a speed of 18 ft/s, is made to slide down a 14 ft chute inclined at $35^\circ.$ The surface of the chute has a coefficient of kinetic friction $\mu_k,$ and at its lower end, it smoothly lets the crate onto a horizontal trajectory. The horizontal surface at the end of the chute has a coefficient of kinetic friction $\mu_k^2.$ Model the crate as a particle, and assume that gravity and the contact forces between the crate and the sliding surface are the only relevant forces. Let $\mu_k = 0.5$ and suppose that once the crate reaches the bottom of the chute and after sliding horizontally for 5 ft, the crate runs into a bumper. If the weight of the crate is W = 110 lb, $\mu_k^2 = 0.33,$ and you model the bumper as a linear spring with constant k and neglect the mass of the bumper, determine the value of k so that the crate comes to a stop 2 ft after impacting with the bumper.

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