Adirondack Savings Bank $($ASB$)$ has $$1$ million in new funds that must be allocated to home loans, personal loans, and automobile loans. The annual rates of return for the three types of loans are $4\%$ for home loans, $11\%$ for personal loans, and $9\%$ for automobile loans. The bank$$s planning committee has decided 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

Max

$\mathrm{.}04$

H $+$

$\mathrm{.}11$

P $+$

$\mathrm{.}09$

A

s$.$t$.$

$1$

H $+$

$0$

P $+$

$0$

A $>=$

$400000$

Minimum Home Loans

$0$

H $+$

$1$

P $+$

$-\mathrm{.}6$

A

$0$

Personal Loan Requirement

$1$

H $+$

$1$

P $+$

$1$

A $=$

$1000000$

Amount of New Funds

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

Loan type Allocation

Home $

$400000$

Personal $

$225000$

Automobile $

$375000$

What is the total annual return?

If required, round your answer to nearest whole dollar amount.

$

$74500$

What is the annual percentage return?

If required, round your answer to two decimal places.

$7\mathrm{.}45$

$\%$

$($c$)$ If the interest rate on home loans increases to $9\%,$ would the amount allocated to each type of loan change?

No

Explain.

An increase to $9\%$

is

within the allowable range, so the amount allocated to each type of loan

would not

change.

$($d$)$ Suppose the total amount of new funds available is increased by $$10,000.$ What effect would this have on the total annual return? Explain.

If required, round your answer to nearest whole dollar amount.

An increase of $$10,000$ to the total amount of funds available would increase the total annual return by $

$.$

$($e$)$ Assume that ASB has the original $$1$ million in new funds available and that the planning committee has agreed to relax the requirement that at least $40\%$ of the new funds must be allocated to home loans by $1\%.$ How much would the annual return change?

If required, round your answer to nearest whole dollar amount.

$

How much would the annual percentage return change?

If required, round your answer to two decimal places.$($b$)$ How much should be allocated to each type of loan?

What is the total annual return?

If required, round your answer to nearest whole dollar amount.

$

What is the annual percentage return?

If required, round your answer to two decimal places.

$\%$