Solve Problems P1.1.5 Insight Through Computing: A MATLAB Introduction to Computational Science and Engineering

P1.1.5 Asolid is a Platonic solid if each face is identical in size and shape. There are only ve: Solid Faces Face Shape Tetrahedron 4 equilateral triangle Cube 6 square Octahedron 8 equilateral triangle Dodecahedron 12 regular pentagon Icosahedron 20 equilateral triangle KeyattributesofagivenPlatonicsolidP includeitsedgelengthE,inradiusr,outradiusR,surface area S, and volume V. The inradius of P is the distance from its center to the centroid of any face. It is the radius of the largest sphere that ts inside P. The outradius of P is the distance from its center to any vertex. It is the radius of the smallest sphere that encloses P. Here is a table that species, for each Platonic solid, the values of r, R, S, and V as a function of E: Solid r R S V Tetrahedron 6 12 E 6 4 E 3E2 2 12 E3 Cube 1 2E 3 2 E 6E2 E3 Octahedron 6 6 E 2 2 E 23E2 2 3 E3 Dodecahedron 250+1105 20 E 15+3 4 E 325+105E2 15+75 4 E3 Icosahedron 42+185 12 E 10+25 4 E 53E2 15+55 12 E3 Using the entries in this table, it is possible to solve various Platonic solid problems. For example, here is a fragment that assigns to Vol20 the volume of the largest icosahedron that ts inside the unit sphere: E = 1/(sqrt(10 + 2*sqrt(5))/4); Vol20 = ((15 + 5*sqrt(5))/4)*E3; Dene T, C, O, D, and I as follows: T: the largest tetrahedron that ts inside the unit sphere C: the largest cube that ts inside T O: the largest octahedron that ts inside C D: the largest dodecahedron that ts inside O I: the largest iscosahedron that ts inside D. Write a script that prints a table showing the inradius, outradius, and edge length of these (nested) Platonicsolids. Thetableshouldhaveaheading,andnumericalvaluesshouldbedisplayedthrough the fteenth decimal place.