#### Answer

Using the Nearest Neighbor Method, the Hamilton circuit is B,D,C,A,B. The total weight of the circuit is 82.

#### 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 B. We can travel along edge BA, BC, or BD. The weight of these paths are 40, 24, and 12 respectively. We should travel along the edge BD because it has the smallest weight.
From vertex D, we can travel along edge DA or DC. The weights of these paths are 14 and 10 respectively. We should travel along edge DC because it has the smallest weight.
From vertex C, the path must travel to vertex A, and then finally back to vertex B to complete the Hamilton circuit.
Using the Nearest Neighbor Method, the Hamilton circuit is B,D,C,A,B. The edges in this path are BD, DC, CA, and AB. The weights of these edges are 12, 10, 20, and 40.
We can find the total weight of this Hamilton circuit.
total weight = 12 + 10 + 20 + 40
total weight = 82
Using the Nearest Neighbor Method, the Hamilton circuit is B,D,C,A,B. The total weight of the circuit is 82.