#### Answer

Using the Nearest Neighbor Method, the Hamilton circuit is A,D,E,C,B,A. The total weight of the circuit is 1457. This means that the total cost is $1457.

#### Work Step by Step

With the Nearest Neighbor Method, we should choose the edge which has the smallest weight for each step along the path.
Let's start at vertex A. We can travel along edge AB, AC, AD, or AE. The weights of these paths are 500, 200, 185, and 205, respectively. We should travel along the edge AD because it has the smallest weight.
From vertex D, we can travel along edge DB, DC, or DE. The weights of these paths are 360, 320, and 302, respectively. We should travel along edge DE because it has the smallest weight.
From vertex E, we can travel along edge EB or EC. The weights of these paths are 340 and 165, respectively. We should travel along edge EC because it has the smallest weight.
From vertex C, the path must travel to vertex B, and then finally back to vertex A to complete the Hamilton circuit.
Using the Nearest Neighbor Method, the Hamilton circuit is A,D,E,C,B,A. The edges in this path are AD, DE, EC, CB, and BA. The weights of these edges are 185, 302, 165, 305, and 500.
We can find the total weight of this Hamilton circuit.
total weight = 185 + 302 + 165 + 305 + 500
total weight = 1457
Using the Nearest Neighbor Method, the Hamilton circuit is A,D,E,C,B,A. The total weight of the circuit is 1457. This means that the total cost is $1457.