A farmer has three hundred acres of arable land on which she wants to plant cauliflower and cabbage. The farmer has $\$ 17,500$ available for planting and $\$ 12,000$ for fertilizer. Planting 1 acre of cauliflower costs $\$ 70$, and planting 1 acre of cabbage costs $\$ 35$. Fertilizer costs $\$ 25$ for 1 acre of cauliflower and $\$ 55$ for 1 acre of cabbage.

(a) Find a system of inequalities that describes the number of acres of each crop that the farmer can plant with the available resources. Graph the feasible region.

(b) Can the farmer plant one hundred fifty-five acres of cauliflower and one hundred fifteen acres of cabbage?

(c) Can the farmer plant one hundred fifteen acres of cauliflower and one hundred seventy-five acres of cabbage?

C-town brewery brews two beers: Expansion Draft and Burning River. Expansion Draft sells for $\$20$ per barrel, while Burning River sells for $\$8$ per barrel. Producing a barrel of Expansion Draft takes 8 pounds of corn and 4 pounds of hops. Producing a barrel of Burning River requires 2 pounds of corn, 6 pounds of rice, and 3 pounds of hops. The brewery has 500 pounds of corn, 300 pounds of rice, and 400 pounds of hops. Assuming a linear relationship, use the Excel Solver to determine the optimal mix of Expansion Draft and Burning River that maximizes C-town’s revenue.

A diet is being prepared for the University of Arizona dorms. The objective is to feed the students at the least cost, but the diet must have between 1,800 and 3,600 calories. No more than 1,400 calories can be starch, and no fewer than 400 can be protein. The varied diet is to be made of two foods: A and B. Food A costs $\$0.75$ per pound and contains 600 calories, 400 of which are protein and 200 starch. No more than 2 pounds of food A can be used per resident. Food B costs $\$0.15$ per pound and contains 900 calories, of which 700 are starch, 100 are protein, and 100 are fat. Write the equations representing this information.

A biologist has two brine solutions, one containing 5% salt and another containing $20 \%$ salt. How many milliliters of each solution should she mix to obtain $1 \mathrm{~L}$ of a solution that contains $14 \%$ salt?

Solve the system of equations.

$$\left\{\begin{array}{l}y=x^2+2 x \\ y=6+x\end{array}\right.$$

A system of linear equations is given. (a) Find the complete solution of the system, or show that there is no solution. (b) State whether the system is inconsistent, dependent, or neither.

$$\left\{\begin{aligned} 2 x-y+z &=0 \\ 3 x+2 y-3 z &=1 \\ x-4 y+5 z &=-1 \end{aligned}\right.$$

Use the substitution method to find all solutions of the system of equations.

$$\left\{\begin{array}{r}x-y=2 \\ 2 x+3 y=9\end{array}\right.$$

Use back-substitution to solve the triangular system.

$$\left\{\begin{aligned} x-3 y+z &=0 \\ y-z &=3 \\ z &=-2 \end{aligned}\right.$$

Use the substitution method to find all solutions of the system of equations.

$$\left\{\begin{array}{r} x+y^2 &=0 \\ 2 x+5 y^2 &=75 \end{array}\right.$$

A system of inequalities and several points are given. Determine which points are solutions of the system.

$$\left\{\begin{array}{l}3 x-2 y \leq 5 \\ 2 x+y \geq 3\end{array} ; \quad(0,0),(1,2),(1,1),(3,1)\right.$$

Write the form of the partial fraction decomposition of the function (as said in the previous example). Do not determine the numerical values of the coefficients.

$$\frac{x^2}{(x-3)\left(x^2+4\right)}$$

A chemist has two large containers of sulfuric acid solution, with different concentrations of acid in each container. Blending $300 \mathrm{~mL}$ of the first solution and $600 \mathrm{~mL}$ of the second gives a mixture that is $15 \%$ acid, whereas blending $100 \mathrm{~mL}$ of the first with $500 \mathrm{~mL}$ of the second gives a $12 \frac{1}{2} \%$ acid mixture. What are the concentrations of sulfuric acid in the original containers?

A farmer has 500 acres of arable land on which he wants to plant potatoes and corn. The farmer has $\$ 40,000$ available for planting and $\$ 30,000$ for fertilizer. Planting 1 acre of potatoes costs $\$ 90$, and planting 1 acre of corn costs $\$ 50$. Fertilizer costs $\$ 30$ for 1 acre of potatoes and $\$ 80$ for 1 acre of corn.

(a) Find a system of inequalities that describes the number of acres of each crop that the farmer can plant with the available resources. Graph the feasible region.

(b) Can the farmer plant $300$ acres of potatoes and $180$ acres of corn?

(c) Can the farmer plant $150$ acres of potatoes and $325$ acres of corn?

A customer in a coffee shop purchases a blend of two coffees: Kenyan, costing $\$ 3.50$ a pound, and Sri Lankan, costing $\$ 5.60$ a pound. He buys $3 \ \mathrm{lb}$ of the blend, which costs him $\$ 11.55$. How many pounds of each kind went into the mixture?