3. $[-/ 3$ Points $]$ DETAILS WANEFMAC8 6.2.043. MY NOTES Investments: Financial Stocks At the end of September 2021, Toronto Dominion Bank (TD) stock cost $\$ 66$ per share, was expected to yield $4 \%$ per year in dividends (calculated on the total value of TD stock you bought), and had a risk index of 3.0 per share, while CNA Financial Corp. (CNA) stock cost $\$ 42$ per share, was expected to yield $3.5 \%$ per year in dividends, and had a risk index of 2.0 per share. $t$ You have up to $\$ 25,800$ to invest in these stocks, and would like to earn at least $\$ 919$ in dividends over th course of a year. (Assume the dividends to be unchanged for the year.) How many shares (to the nearest tenth of a unit) of each stock should you purchase to meet your requirements and minimize the total risk index for your portfolio? Toronto Dominion Bank shares CNA Financial Corp. shares What is the minimum total risk index? (Round your answer to two decimal places.) Show My Work (Optiona) ?

Step 1 ：Define the problem as a linear programming problem. The objective is to minimize the total risk index, which is the sum of the risk indices of the stocks purchased. The risk index of each stock is given by the number of shares purchased times the risk index per share. Therefore, the objective function to be minimized is \(3x + 2y\), where \(x\) is the number of Toronto Dominion Bank shares and \(y\) is the number of CNA Financial Corp. shares.

Step 2 ：There are two constraints. The first constraint is that the total cost of the stocks purchased cannot exceed $25,800. The cost of each stock is given by the number of shares purchased times the cost per share. Therefore, the first constraint is \(66x + 42y \leq 25800\).

Step 3 ：The second constraint is that the total dividends earned must be at least $919. The dividends from each stock are given by the number of shares purchased times the dividend yield per share. Therefore, the second constraint is \(0.04*66x + 0.035*42y \geq 919\).

Step 4 ：Since the number of shares cannot be negative, there are also non-negativity constraints: \(x \geq 0\) and \(y \geq 0\).

Step 5 ：Solve the linear programming problem using a suitable method. The optimal solution is \(x = 348.1\) and \(y = 0\).

Step 6 ：Substitute the optimal solution into the objective function to find the minimum total risk index. The minimum total risk index is \(3*348.1 + 2*0 = 1044.32\).

Step 7 ：Final Answer: \(\boxed{348.1, 0, 1044.32}\)