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1)Donatello starts with a marble cube of side length \$10.\$ He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length \$s.\$ Find \$s.\$

[asy] import three; size(7cm); unitsize(1 cm); currentprojection=perspective(6,3,2); triple A, B, C, D, E, F, G, H; A = (1,1,0); B = (1,0,0); C = (0,0,0); D = (0,1,0); E = (1,1,1); F = (1,0,1); G = (0,0,1); H = (0,1,1); draw(A--B--F--G--H--D--cycle); draw(E--A); draw(E--F); draw(E--H); draw((2*F + B)/3--(2*F + E)/3--(2*F + G)/3); draw((2*E + A)/3--(2*E + F)/3--(2*E + H)/3--cycle); draw((2*A + B)/3--(2*A + E)/3--(2*A + D)/3); draw((2*D + A)/3--(2*D + H)/3); draw((2*B + A)/3--(2*B + F)/3); draw((2*H + D)/3--(2*H + E)/3--(2*H + G)/3); draw((2*G + F)/3--(2*G + H)/3); //dot("\$C\$", E, SW); //dot("\$X\$", (2*E + H)/3, SE); //dot("\$Y\$", (2*E + F)/3, SW); //dot("\$Z\$", (2*E + A)/3, SW); [/asy]

2)Let \$ABCDEFGH\$ be a rectangular prism, where \$EF = 4\$, \$EH = 5\$, and \$EA = 6\$. Find the volume of pyramid \$CFAH\$.

[asy] unitsize(1 cm); pair A, B, C, D, E, F, G, H; A = (0,0); B = (0,2); C = (0,2) + 2*dir(-30); D = 2*dir(-30); E = A + 3*dir(20); F = B + 3*dir(20); G = C + 3*dir(20); H = D + 3*dir(20); fill(A--H--F--cycle,gray(0.8)); draw(A--B--F--G--H--D--cycle); draw(C--B); draw(C--G); draw(C--D); draw(E--A,dashed); draw(E--F,dashed); draw(E--H,dashed); draw(C--A); draw(C--F); draw(C--H); draw(A--H--F--cycle,dashed); label("\$A\$", A, SW); label("\$B\$", B, NW); label("\$C\$", C, N); label("\$D\$", D, S); label("\$E\$", E, S); label("\$F\$", F, N); label("\$G\$", G, NE); label("\$H\$", H, SE); [/asy]

3)Let \$A_1 A_2 A_3 A_4\$ be a regular tetrahedron. Let \$P_1\$ be the center of face \$A_2 A_3 A_4,\$ and define vertices \$P_2,\$ \$P_3,\$ and \$P_4\$ the same way. Find the ratio of the volume of tetrahedron \$A_1 A_2 A_3 A_4\$ to the volume of tetrahedron \$P_1 P_2 P_3 P_4.\$

[asy] import three; size(7cm); unitsize(1 cm); currentprojection=perspective(6,3,2); triple A, B, C, D, E, F, G, H; real ang = 40; A = (Cos(ang), Sin(ang), 0); B = (Cos(ang + 120), Sin(ang + 120), 0); C = (Cos(ang + 240), Sin(ang + 240), 0); D = (0,0,sqrt(2)); E = (B + C + D)/3; F = (A + C + D)/3; G = (A + B + D)/3; H = (A + B + C)/3; draw(B--A--C--D--cycle); draw(A--D); draw(B--C,dashed); draw(E--F--G--H--cycle,dashed); draw(E--G,dashed); draw(F--H,dashed); label("\$A_1\$", A, S); label("\$A_2\$", B, dir(0)); label("\$A_3\$", C, W); label("\$A_4\$", D, N); dot("\$P_1\$", E, N); dot("\$P_2\$", F, W); dot("\$P_3\$", G, dir(0)); dot("\$P_4\$", H, S); [/asy]

Note: A tetrahedron is called regular if all the edges lengths are equal, so all the faces are equilateral triangles.

Feb 20, 2020