$(4$ pts$)$ The size of the pores that exist in a green body ceramic depends on the arrangement and size of the powder particles in the compact. We will examine this using a simple two$-$dimensional model of particle packing.

Below is a two$-$dimensional simple cubic arrangement of spheres with radius $R.$

$(1)$ Assume a spherical pore as shown in the schematic above. Calculate the size of the pore $($which would have a radius $r)$ that would exist between these four powder particles that have particle sizes of $150m.$

$(2)$ Determine what is the maximum size for a spherical guest particle to be filled in the center pore of the cube. What is the radius ratio of this guest particle vs$.$ the initial sphere?