Adirondack Savings Bank $($ASB$)$ has $$1$ million in new funds that must be allocated to home loans, personal loans, and automobile loans. The annual rates of return for the three types of loans are $7\%$ for home loans, $14\%$ for personal loans, and $8\%$ for automobile loans. The bank's planning committee has decided

$4.$ that at least $40\%$ of the new funds must be allocated to home loans. In addition, the planning committee has specified that the amount allocated to personal loans cannot exceed $60\%$ of the amount allocated to automobile loans.

$($a$)$ Formulate a linear programming model that can be used to determine the amount of funds ASB should allocate to each type of loan to maximize the total annual return for the new funds. If the constant is $"1"$ it must be entered in the box. If your answer is zero enter $"0".$

Let $H=$ amount allocated to home loans

$P=$ amount allocated to personal loans

$A=$ amount allocated to automobile loans

$($b$)$ How much should be allocated to each type of loan?

$\backslash$table$[[$Loan type,Allocation$],[$Home$,$$