Exam 6: Systems of Equations and Inequalities

An accounting firm charges $2500 for an audit and $350 for a tax return. Research and available resources have indicated the following constraints. The firm has 900 hours of staff time available each week. The firm has 155 hours of review time available each week. Each audit requires 75 hours of staff time and 10 hours of review time. Each tax return requires 12.5 hours of staff time and 2.5 hours of review time. The accounting firm lowers its charge for an audit to $2000. What numbers of audits and tax returns will bring in an optimal revenue?

Solve the system of linear equations. $\left\{ \begin{array} { c } x + y + z + w = 0 \\- 3 x - 2 y + z - w = - 5 \\- 5 x + 4 z - w = - 20 \\- 5 x + y + z + 4 w = - 15\end{array} \right.$

A chemist needs 8 liters of a 45% acid solution. The solution is to be mixed from three solutions whose acid concentrations are 10%, 20%, and 50%. How many liters of the 50% solution should the chemist use if trying to use as little as possible of the 50% solution?

Solve the system by the method of substitution. $\left\{ \begin{array} { c } x - y = - 7 \\x ^ { 2 } - y = - 5\end{array} \right.$

Solve the system of equations below: $8 x - 8 y - 8 z = - 24$ $2 x + 2 y - 2 z = 2$ $x - 2 y - z = - 5$

Write a system of inequalities whose solution set is graphed below.

Sketch the graph of the inequality below. $y > - x - 3$

A residential building contractor borrowed $24,000 to complete a new home. Some of the money was borrowed at 5%, some at 7%, and some at 8%. How much was borrowed at each rate if the annual interest owed was $1480 and the amount borrowed at 7% is two times more than the amount borrowed at 8%?

A fruit juice company makes two drinks by blending apple and pineapple juices. The percent of apple juice and pineapple juice in each drink are shown in the table. Mixture Drink A Drink B Apple juice 30\% 60\% Pineapple juice 70\% 40\% There are 1000 liters of apple and 1500 liters of pineapple juice available. The profit for drink A is $0.70 per liter and the profit for drink B is $0.60 per liter. What is the optimal production level for each type of drink? What is the optimal profit?

A dietitian designs a special diet supplement using two different foods. Each ounce of food X contains 15 units of calcium, 10 units of iron, and 20 units of vitamin B. Each ounce of food Y contains 20 units of calcium, 10 units of iron, and 15 units of vitamin B. The minimum daily requirements for the diet are 360 units of calcium, 320 units of iron, and 370 units of vitamin B. Which combinations of foods X and Y below, if any, can be given to the patient to meet the minimum daily requirements? Combination I : 13 ounces of food $\mathrm { X }$ and 16 ounces of food $\mathrm { Y }$ Combination II : 11 ounces of food $X$ and 20 ounces of food $Y$ Combination II : 20 ounces of food $X$ and 23 ounces of food $Y$

Solve the system of equations below: $- 7 x + 6 y + 5 z = 0$ $- 2 x - 2 y - 2 z = 0$ $2 x - 7 y - 3 z = 0$

A total of $29,000 is invested in two corporate bonds that pay 12% and 10.5% simple interest. The total annual interest is $3195. How much is invested in the 10.5% bond?

A humanitarian agency can use two models of vehicles for a refugee rescue mission. Each model A vehicle costs $1000 and each model B vehicle costs $1500. Mission strategies and objectives indicate the following constraints. A total of at least 20 vehicles must be used. A model A vehicle can hold 45 boxes of supplies.A model B vehicle can hold 30 boxes of supplies. The agency must deliver at least 690 boxes of supplies to the refugee camp. A model A vehicle can hold 20 refugees. A model B vehicle can hold 32 refugees. The agency must rescue at least 520 refugees. What is the optimal number of vehicles of each model that should be used? What is the optimal cost?

You invest $2500 in a fishing lure business. A lure costs $1.47 to produce and will be sold for $4.97. How many lures must you sell to break even?

Graph the solution set of the system of inequalities below. $\left\{ \begin{aligned}x - y & \leq 2 \\- x - y & \leq 2 \\- y & \geq 0\end{aligned} \right.$

Write an inequality for the shaded region shown in the figure.

Determine which ordered pair is a solution of the system. $\left\{ \begin{aligned}6 x - 6 y & = - 9 \\- 2 x - 7 y & = 2\end{aligned} \right.$

Sketch the graph and label the vertices of the solution set of the system of inequalities. Shade the solution set. $\left\{ \begin{array} { l } x ^ { 2 } + y \leq 4 \\x \geq 1 \\y \geq 0\end{array} \right.$

Find the equation of the parabola $y = a x ^ { 2 } + b x + c$ that passes through the points. $( 0,2 ) , ( 1,2 ) , ( 2,4 )$

The concentration $C$ (in parts per million) of carbon dioxide in the atmosphere is measured at the Mauna Loa Observatory in Hawaii. The greatest monthly carbon dioxide concentration for each year from 2002 to 2006 is shown in the table. Year t Concentration, C 2002 0 375.55 2003 1 378.35 2004 2 380.63 2005 3 382.26 2006 4 384.92 Solve the following system for $a$ and $b$ to find the least squares regression line $y = a t + b$ for the data. Let $t$ represent the year, with $t = 0$ corresponding to 2002. $\left\{ \begin{array} { r } 5 b + 10 a = 1901.71 \\10 b + 30 a = 3826.07\end{array} \right.$ Use the least square regression line to predict the largest monthly carbon dioxide concentration in 2011. Round your answer to the nearest hundredths part per million.