The amount of pressure required to sink a large, heavy object in a soft homogeneous soil that lies above a hard base soil can be predicted by the amount of pressure required to sink smaller objects in the same soil. Specifically, the amount of pressure p required to sink a circular plate of radius r a distance d in the soft soil, where the hard base soil lies a distance D > d below the surface, can be approximated by an equation of the form
p = k1ek2r + k3r,
where k1, k2, and k3 are constants, with k2 > 0, depending on d and the consistency of the soil but not on the radius of the plate.
a. Find the values of k1, k2, and k3 if we assume that a plate of radius 1 in. requires a pressure of 10 lb/in.2 to sink 1 ft in a muddy field, a plate of radius 2 in. requires a pressure of 12 lb/in.2 to sink 1 ft, and a plate of radius 3 in. requires a pressure of 15 lb/in.2 to sink this distance (assuming that the mud is more than 1 ft deep).
b. Use your calculations from part (a) to predict the minimal size of circular plate that would be required to sustain a load of 500 lb on this field with sinkage of less than 1 ft.