ANSWERS ONLY

Construct the complementary graph (draw the actual graph) of the following graph representation. Use vertex labels a,b,c,d,e,f when constructing your graph. O O O HO 0 O 10 1 0 0 1 HOOHOP OHOI OHH Attach the drawing/photo/pdf to this question as your answer.A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x, y, z) = 25 - 5(x2 + y + z?) C. Use the fact that heat flow is given by the vector field F = -KVw and the rate of heat flow across a surface S within the solid is given by -K JIs Vwas. Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper (K = 400 kW/(m . K)). (Use symbolic notation and fractions where needed.) -K Vwas = kW SA varying current i(t) = t(16 - t) A (t in seconds) flows through a long straight wire that lies along the x-axis. The current produces a magnetic field B whose magnitude at a distance r from the wire is B = , T. Furthermore, at the point P, B points away from the observer as shown in the figure. Wire loop C Rectangular region R P = (x, y) B O Volt meter x=0 Calculate the flux D(t), at time t, of B through a rectangle of dimensions L X H = 9 x 2 m whose top and bottom edges are parallel to the wire and whose bottom edge is located d = 0.5 m above the wire. Assume that the rectangle and the wire are located in the same plane. (Use symbolic notation and fractions where needed. Let I = i(t) and express your answer in terms of Mo and I.) D(1) = T . m2 Use Faraday's Law to determine the voltage drop around the rectangular loop (the boundary of the rectangle) at time t = 7. Assume Mo = 4x . 10- T . m/A. (Use symbolic notation and fractions where needed.) E . dr = V CLet T be the triangular region region with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1) oriented with upward-pointing normal vector. Assume distances are in meters. N (0, 0, 1) (0, 1, 0) (1, 0, 0) A fluid flows with constant velocity field v = 16k (meters per second). (a) Calculate the flow rate through T. (Give your answer as a whole or exact number.) / v . as = m' /s (b) Calculate the flow rate through D, the projection of Tonto the xy-plane [the triangle with verticies (0, 0, 0), (1, 0, 0), (0, 1, 0)]. (Give your answer as a whole or exact number.) v . as = m'/sLet er = (*, -, 2 be a unit radial vector, where r = Vx2 + yz + z. (a) Calculate the integral of F = ever over the upper hemisphere of x + y + z = 81 with the normal pointing outward. (Give your answer in exact form. Use symbolic notation and fractions where needed.) F . dS = S (b) Calculate the integral of F = 9e 'er over the octant x 2 0, y 2 0, z 2 0 of the unit sphere centered at the origin. (Give your answer in exact form. Use symbolic notation and fractions where needed.) F . dS =