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Graded AssignmentMTH203B/204B Geometry | Unit 6 | Lesson 16: Beyond Euclidean Geometry Unit Test

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Graded AssignmentUnit Test, Part 2Answer the questions. When you are finished, submit this test to your teacher by the due date for full credit.(18 points)

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1. The graph below represents five cities and the routes between them. An out-of-state road inspector is asked to fly into Pottsville (P). From Pottsville, the inspector must inspect each road on the map, making only one trip along each road, and then fly out of Tinkertown (T).

a.) Will the road inspector be able to complete the inspection as described? Mathematically explain why or why not and, if such an inspection is possible, describe the route.2 vertices P and T have an odd degree and all other vertices have an even degree, which satisfies the criteria for Eulers path. Eulers path sates that every edge can be crossed just once. The below mentioned path is one example.

P R C M T C P T covers b.) The inspector would prefer to fly into and out of the same town, but still drive each road only once. Would that be possible? Mathematically explain why or why not and, if such an inspection is possible, describe the route.Since all the vertices are not even, Eulers circuit, which starts and ends at the same point and crossing every edge just once, is not possible according to the criteria for Eulers circuit.c.) Below, the map has been expanded to show additional cities and routes. A trucker who lives in Jonesboro (J) needs to make deliveries in each town and then return home without passing through any of the towns more than once. What type of mathematical circuit is the trucker hoping to use? If the trucker can complete the circuit, describe the route.The truck driver needs to use the Hamiltonian Circuit.

One possible route is: J R A V C M T P J.

(10 points)

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2. Compare and contrast Euclidean geometry and spherical geometry. Be sure to include these points:a.) Describe the role of the Parallel Postulate in spherical geometry.b.) How are triangles different in spherical geometry as opposed to Euclidean geometry?c.) Geodesicsd.) Applications of spherical geometryAnswer:

a) Spherical geometry negates parallel postulate, since it does not deal with lines, but deals with points and circles and there cannot be two great circles that do not intersect each other.

On the other hand Euclidean geometry deals with points and lines and hence in its 5th postulate states that given a line and a point that is not on that line, there is only one line that contains that point and is parallel to the given line.b) In Euclidean geometry, three points that are non coplanar, when joined together form a triangle. Whereas in spherical geometry, any three points that do not share the same great circle are joined with arcs that run along the great circles as sides to form a spherical triangle.c) Geodesic is the shortest distance between two points in Euclidean geometry, which is a straight line.

In spherical geometry geodesics is a great circle, which is a circle on the surface of a sphere, whose center coincides with the center of the sphere and has a diameter which is the diameter of the spheres

d) Spherical geometry is applied in the aviation industry to find the shortest distances between places on earth, which is spherical in shape. Great circles can be used to find the shortest distances. Shortest distances lie along the great circle.(7 points)

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3. Use your knowledge of computer logic to answer these questions.a.) 10001 base 2 = ______ base 10b.) 1101101 base 2 = _____ base 10c.) What type of gate does this input-output table correspond to?InputOutput

AB?

0

0

1

11

0

0

11

0

1

1

d.) The expression that describes the network of logic gates:

Is [A AND (NOT B)] OR (NOT B). Complete the input-output table for the network:Answer:

a) 124 + 120=17b) 126 + 125 + 123 + 122 + 120 = 64 + 32 + 8 + 4 + 1 = 109c) OR gate

d) NOT B

Your Score___ of 35

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