A fast-food chain plans to expand by opening several new restaurants. The chain operates two types of restaurants, drive-through, and full-service. A drive-through restaurant costs $\$ 100,000$ to construct, requires $5$ employees, and has an expected annual revenue of $\$ 200,000$. A full-service restaurant costs $\$ 150,000$ to construct, requires $15$ employees, and has an expected annual revenue of $\$ 500,000$. The chain has $\$ 2,400,000$ in capital available for expansion. Labor contracts require that they hire no more than $210$ employees, and licensing restrictions require that they open no more than $20$ new restaurants. How many restaurants of each type should the chain open in order to maximize the expected revenue? What is the maximum expected revenue? How much of their capital will they use and how many employees will they hire?

An investor has $24,000 to invest in bonds of AAA and B qualities. The AAA bonds yield an average of 6%, and the B bonds yield 10%. The investor requires that at least three times as much money should be invested in AAA bonds as in B bonds. How much should be invested in each type of bond to maximize the return? What is the maximize return?

Ella's cat has climbed up onto the roof of her house and cannot get down! The roof of the house is 24 feet above the ground. The fire department has set up a 28-foot ladder to climb to rescue the cat. The base of the ladder is resting on the ground, and the top of the ladder extends 2 feet beyond the edge of the roof. How far away from the base of the house is the base of the ladder? You may want to draw a diagram to help you solve the problem.

A small company manufactures three different electronic components for computers. Component A requires 2 hours of fabrication and 1 hour of assembly; component B requires 3 hours of fabrication and 1 hour of assembly; and component C requires 2 hours of fabrication and 2 hours of assembly. The company has up to 1,000 labor-hours of fabrication time and 800 labor-hours of assembly time available per week. The profit on each component, A, B, and C, is $7,$8, and $10, respectively. How many components of each type should the company manufacture each week in order to maximize it s profit (assuming that all components manufactured can be sold)? What is the maximum profit?

An investor has $24,000 to invest in bonds of AAA and B qualities. The AAA bonds yield an average of 6%, and the B bonds yield 10%. The investor requires that at least three times as much money should be invested in AAA bonds as in B bonds. How much should be invested in each type of bond to maximize the return? What is the maximum return?

An investor has $\$ 24,000$ to invest in bonds of AAA and B qualities. The AAA bonds yield an average of $6 \%$, and the $B$ bonds yield $10 \%$. Does the investor require that at least three times as much money should be invested in AAA bonds as in $B$ to maximize the return? What is the maximum return?

An electronics firm manufactures two types of personal computers - a standard model and a portable model. The production of a standard computer requires a capital expenditure of $\$ 400$ and $40$ hours of labor. The production of a portable computer requires a capital expenditure of $\$ 250$ and $30$ hours of labor. The firm has $\$ 20,000$ capital and $2,160$ labor hours available for the production of standard and portable computers. (A) What is the maximum number of computers the company is capable of producing? (B) If each standard computer contributes a profit of $\$ 320$ and each portable model contributes a profit of $\$ 220$, how much profit will the company make by producing the maximum number of computers determined in part (A)? Is this the maximum profit? If not, what is the maximum profit?

An electronics firm manufactures two types of personal computers-a standard model and a portable model. The production of a standard computer requires a capital expenditure of $400 and 40 hours of labor. The production of a portable computer requires a capital expenditure of$250 and 30 hours of labor. The firm has $20,000 capital and 2,160 labor-hours available for production of standard and portable computers. (A) What is the maximum number of computers the company is capable of producing? (B) If each standard computer contributes a profit of$320 and each portable model contributes a profit of $220, how much profit will the company make by producing the maximum number of computers determined in part (A)? Is this the maximum profit? If not, what is the maximum profit?

A manufacturing company makes two types of water skis, a trick ski, and a slalom ski. The trick ski requires $6$ labor hours for fabricating and $1$ labor hour for finishing. The slalom ski requires $4$ labor hours for fabricating and $1$ labor hour for finishing. The maximum labor hours available per day for fabricating and finishing are $108$ and $24$, respectively. If $x$ is the number of trick skis and $y$ is the number of slalom skis produced per day, write a system of linear inequalities that indicates appropriate restraints on $x$ and $y$. Find the set of feasible solutions graphically for the number of each type of ski that can be produced.

Because of new federal regulations on pollution, a chemical plant introduced a new, more expensive process to supplement or replace an older process used in the production of a particular chemical. The older process emitted $20$ grams of sulfur dioxide and $40$ grams of particulate matter into the atmosphere for each gallon of chemically produced. The new process emits 5 grams of sulfur dioxide and $20$ grams of particulate matter for each gallon produced. The company makes a profit of $60 \varnothing$ per gallon and $20 \varnothing$ per gallon on the old and new processes, respectively. (A) If the government allows the plant to emit no more than $16,000$ grams of sulfur dioxide and $30,000$ grams of the particulate matter daily, how many gallons of the chemical should be produced by each process to maximize daily profit? What is the maximum daily profit? (B) Discuss the effect on the production schedule and the maximum profit if the government decides to restrict emissions of sulfur dioxide to $11,500$ grams daily and all other data remain unchanged. (C) Discuss the effect on the production schedule and the maximum profit if the government decides to restrict emissions of sulfur dioxide to $7,200$ grams daily and all other data remain unchanged

Why did Lincoln's election prompt the secession of the southern states?

Explain why northerners favored the Wilmot Proviso and why southerners did not.

A farmer can buy two types of plant food, mix $A$ and mix $B$. Each cubic yard of mix $A$ contains $20$ pounds of phosphoric acid, $30$ pounds of nitrogen, and $5$ pounds of potash. Each cubic yard of mix $B$ contains $10$ pounds of phosphoric acid, $30$ pounds of nitrogen, and $10$ pounds of potash. The minimum monthly requirements are $460$ pounds of phosphoric acid, $960$ pounds of nitrogen, and $220$ pounds of potash. If $x$ is the number of cubic yards of mix $A$ used and $y$ is the number of cubic yards of mix $B$ used, write a system of linear inequalities that indicates appropriate restraints on $x$ and $y$. Find the set of feasible solutions graphically for the amounts of mix $A$ and mix $B$ that can be used.

A dietitian in a hospital is to arrange a special diet using two foods. Each ounce of food M contains 30 units of calcium, 10 units of iron, and 10 units of vitamin A. Each ounce of food N contains 10 units of calcium, 10 units of iron, and 30 units of vitamin A. The minimum requirements in the diet are 360 units of calcium, 160 units of iron, and 240 units of vitamin A. If x is the number of ounces of food M used and y is the number of ounces of food N used, write a system of linear inequalities that reflects the conditions indicated. Find the set of feasible solutions graphically for the amount of each kind of food that can be used.