#### Answer

The smallest total weight is 1445 when the Hamilton circuit is A,B,D,C,A or A,C,D,B,A. Since the total weight shows the cost in dollars, the smallest total cost is $1445

#### Work Step by Step

We need to find the Hamilton circuit with the smallest total weight. We can use the Brute Force Method to find the optimal solution. The Brute Force Method involves listing all the possible Hamilton circuits and calculating the total weight of each Hamilton circuit.
To find the total weight of a path, we can simply add up the weight of each edge in the path.
The Hamilton circuit is A,B,C,D,A.
total weight = 460 + 720 + 105 + 210 = 1495
The Hamilton circuit is A,B,D,C,A.
total weight = 460 + 680 + 105 + 200 = 1445
The Hamilton circuit is A,C,B,D,A.
total weight = 200 + 720 + 680 + 210 = 1810
The Hamilton circuit is A,C,D,B,A.
total weight = 200 + 105 + 680 + 460 = 1445
The Hamilton circuit is A,D,B,C,A.
total weight = 210 + 680 + 720 + 200 = 1810
The Hamilton circuit is A,D,C,B,A.
total weight = 210 + 105 + 720 + 460 = 1495
Using the Brute Force Method, we can see that the smallest total weight is 1445 when the Hamilton circuit is A,B,D,C,A or A,C,D,B,A. Since the total weight shows the cost in dollars, the smallest total cost is $1445