Perth Mining Company operates two mines for the purpose of extracting gold and silver. The Saddle Mine costs $\$ 14,000 /$ day to operate, and it yields $50 \mathrm{oz}$ of gold and $3000 \mathrm{oz}$ of silver per day. The Horseshoe Mine costs $\$ 16,000 /$ day to operate, and it yields $75 \mathrm{oz}$ of gold and 1000 ounces of silver per day. Company management has set a target of at least $650 \mathrm{oz}$ of gold and 18,000 oz of silver.

a. How many days should each mine be operated so that the target can be met at a minimum cost?

b. Find the range of values that the Saddle Mine's daily operating cost can assume without changing the optimal solution.

c. Find the range of values that the requirement for gold can assume.

d. Find the shadow price for the requirement for gold.

In the following exercise, solve each linear programming problem by the method of corners.

$$\begin{array}{lr} \text { Maximize } & P=x+3 y \\ \text { subject to } & 2 x+y \leq 6 \\ & x+y \leq 4 \\ & x \leq 1 \\ & x \geq 0, y \geq 0 \end{array}$$

When reducing [A|I], a row in the A part consists of all zeros. What does this indicate about $A^{-1}$?

Acoustical Company manufactures a DVD storage cabinet that can be bought fully assembled or as a kit. Each cabinet is processed in the fabrication department and the assembly department. If the fabrication department manufactures only fully assembled cabinets, it can produce 200 units/day; and if it manufactures only kits, it can produce 200 units/day. If the assembly department produces only fully assembled cabinets, it can produce 100 units/day; but if it produces only kits, then it can produce 300 units/day. Each fully assembled cabinet contributes $\$ 50$ to the profits of the company, whereas each kit contributes $\$ 40$ to its profits. How many fully assembled units and how many kits should the company produce per day to maximize its profits?

Formulate but do not solve the problem.

The Johnson Farm has $500$ acres of land allotted for cultivating corn and wheat. The cost of cultivating corn and wheat (including seeds and labor) is $\$ 42$ and $\$ 30$ per acre, respectively. Jacob Johnson has $\$ 18,600$ available for cultivating these crops. If he wishes to use all the allotted land and his entire budget for cultivating these two crops, how many acres of each crop should he plant?

In the following exercise, use the method of corners to solve the linear programming problem.

Find the maximum and minimum values of $Q=3 x+4 y$ subject to

$$\begin{aligned} x-y & \geq-10 \\ x+3 y & \geq 30 \\ 7 x+4 y & \leq 140 \\ x \geq 0, y & \geq 0 \end{aligned}$$

A farmer plans to plant two crops, A and B. The cost of cultivating Crop A is $40/acre whereas that of Crop B is$60/acre. The farmer has a maximum of $\$ 7400$ available for land cultivation. Each acre of Crop $A$ requires 20 labor-hours, and each acre of Crop $B$ requires 25 labor-hours. The farmer has a maximum of 3300 labor-hours available. If he expects to make a profit of $\$ 150$ /acre on Crop $A$ and $\$ 200$ /acre on Crop $B$, how many acres of each crop should he plant to maximize his profit?

a. Find the range of values that the contribution to the profit of an acre of Crop A can assume without changing the optimal solution.

b. Find the range of values that the resource associated with the constraint on the available land can assume.

c. Find the shadow price for the resource associated with the constraint on the available land.

Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type A vessel has 60 deluxe cabins and 160 standard cabins, whereas a type B vessel has 80 deluxe cabins and 120 standard cabins. Under a charter agreement with Odyssey Travel Agency, Deluxe River Cruises is to provide Odyssey with a minimum of 360 deluxe and 680 standard cabins for their 15-day cruise in May. It costs $\$ 44,000$ to operate a type A vessel and $\$ 54,000$ to operate a type B vessel for that period. How many of each type vessel should be used to keep the operating costs to a minimum?

In the following exercise, solve each linear programming problem by the method of corners.

$$\begin{array}{lc} \text { Maximize } & P=3 x-4 y \\ \text { subject to } & x+3 y \leq 15 \\ & 4 x+y \leq 16 \\ & x \geq 0, y \geq 0 \end{array}$$

In the following exercise, use the method of corners to solve the linear programming problem.

$$\begin{array}{lr} \text { Minimize } & C=4 x+y \\ \text { subject to } & 6 x+y \geq 18 \\ & 2 x+y \geq 10 \\ & x+4 y \geq 12 \\ & x \geq 0, y \geq 0 \end{array}$$

Multiple Choice: If A and B are square matrices with AB=I and BA=I , then (A) B is the inverse of A. (B) A and B must be equal. (C) A and B must both be singular. (D) At least one of A and B is singular.

In the following exercise, solve each linear programming problem by the method of corners.

$$\begin{array}{lr} \text { Maximize } & P=x+8 y \\ \text { subject to } & x+y \leq 8 \\ & 2 x+y \leq 10 \\ & x \geq 0, y \geq 0 \end{array}$$

In the following exercise, use the method of corners to solve the linear programming problem.

$$\begin{array}{lr} \text { Minimize } & C=4 x+y \\ \text { subject to } & 6 x+y \geq 18 \\ & 2 x+y \geq 10 \\ & x+4 y \geq 12 \\ & x \geq 0, y \geq 0 \end{array}$$

A company manufactures two products, A and B, on Machines I and II. The company will realize a profit of $\$ 3 /$ unit of Product $A$ and a profit of $\$ 4$ unit of Product $B$. Manufacturing 1 unit of Product $A$ requires $6 \mathrm{~min}$ on Machine I and $5 \mathrm{~min}$ on Machine II. Manufacturing 1 unit of Product $B$ requires $9 \mathrm{~min}$ on Machine I and $4 \mathrm{~min}$ on Machine II. There are $5 \mathrm{hr}$ of time available on Machine I and $3 \mathrm{hr}$ of time available on Machine II in each work shift.

a. How many units of each product should be produced in each shift to maximize the company's profit?

b. Find the range of values that the contribution to the profit of 1 unit of Product $A$ can assume without changing the optimal solution.

c. Find the range of values that the resource associated with the time constraint on Machine I can assume.

d. Find the shadow price for the resource associated with the time constraint on Machine I.