91Ó°ÊÓ

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function $$ z=6 x+10 y $$ Constraints $$ \left\\{\begin{array}{l} x+y \leq 12 \\ x+2 y \leq 20 \\ x \geq 0 \\ y \geq 0 \end{array}\right\\} \begin{aligned} &\text { Quadrant I and } \\ &\text { its boundary } \end{aligned} $$

Graph each linear inequality. \(y \leq 3 x+2\)

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is \(\$ 125\) for the rearprojection televisions and \(\$ 200\) for the plasma televisions. a. Let \(x=\) the number of rear-projection televisions manufactured in a month and \(y=\) the number of plasma televisions manufactured in a month. Write the objective function that describes the total monthly profit. b. The manufacturer is bound by the following constraints: \- Equipment in the factory allows for making at most 450 rear-projection televisions in one month. \- Equipment in the factory allows for making at most 200 plasma televisions in one month. \- The cost to the manufacturer per unit is \(\$ 600\) for the rear-projection televisions and \(\$ 900\) for the plasma televisions. Total monthly costs cannot exceed \(\$ 360,000 .\) Write a system of three inequalities that describes these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) must both be nonnegative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at \((0,0),(0,200),(300,200)\), \((450,100)\), and \((450,0)\).] e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit by manufacturing rear-projection televisions each month and plasma televisions each month. The maximum monthly profit is $\$$

You are about to take a test that contains computation problems worth 6 points each and word problems worth 10 points each. You can do a computation problem in 2 minutes and a word problem in 4 minutes. You have 40 minutes to take the test and may answer no more than 12 problems. Assuming you answer all the problems attempted correctly, how many of each type of problem must you answer to maximize your score? What is the maximum score?

A theater is presenting a program on drinking and driving for students and their parents. The proceeds will be donated to a local alcohol information center. Admission is \(\$ 2\) for parents and \(\$ 1\) for students. However, the situation has two constraints: The theater can hold no more than 150 people and every two parents must bring at least one student. How many parents and students should attend to raise the maximum amount of money?

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & -3 \\ \hline 1 & 2 \\ \hline 2 & 7 \\ \hline 3 & 12 \\ \hline 4 & 17 \\ \hline \end{array} $$

On June 24,1948 , the former Soviet Union blocked all land and water routes through East Germany to Berlin. A gigantic airlift was organized using American and British planes to bring food, clothing, and other supplies to the more than 2 million people in West Berlin. The cargo capacity was 30,000 cubic feet for an American plane and 20,000 cubic feet for a British plane. To break the Soviet blockade, the Western Allies had to maximize cargo capacity, but were subject to the following restrictions: \- No more than 44 planes could be used. \- The larger American planes required 16 personnel per flight, double that of the requirement for the British planes. The total number of personnel available could not exceed 512 . \- The cost of an American flight was \(\$ 9000\) and the cost of a British flight was \(\$ 5000\). Total weekly costs could not exceed \(\$ 300,000\). Find the number of American and British planes that were used to maximize cargo capacity.

Graph each linear inequality. \(x \leq-4\)

a. Create a scatter plot for the data in each table. b. Use the shape of the scatter plot to determine if the data are best modeled by a linear function, an exponential function, a logarithmic function, or a quadratic function. $$ \begin{array}{|c|r|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & -4 \\ \hline 1 & -1 \\ \hline 2 & 0 \\ \hline 3 & -1 \\ \hline 4 & -4 \\ \hline \end{array} $$

Graph each linear inequality. \(y>-2\)