(a) In Problem 48, let s(t) be the distance measured down the inclined plane from th...

(a) In Problem 48, let s(t) be the distance measured down the inclined plane from the highest point. Use ds/dt = v(t) and the solution for each of the three cases in part (b) of Problem 48 to find the time that it takes the box to slide completely down the inclined plane. A root-finding application of a CAS may be useful here.

(b) In the case in which there is friction ( μ ≠ 0) but no air resistance, explain why the box will not slide down the plane starting from rest from the highest point above ground when the inclination angle θ satisfies tan θ ≤ μ.

(c) The box will slide downward on the plane when tan θ ≤ μ if it is given an initial velocity v(0) = v₀ > 0. Suppose that and θ = 23°. Verify that tan θ ≤ μ. How far will the box slide down the plane if v₀ = 1 ft /s?

(d) Using the values approximate the smallest initial velocity v₀ that can be given to the box so that, starting at the highest point 50 ft above ground, it will slide completely down the inclined plane. Then find the corresponding time it takes to slide down the plane.