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 \(\quad z=4 x+y\) Constraints \(\quad x \geq 0, y \geq 0\) \(2 x+3 y \leq 12\) \(x+y \geq 3\)

A television manufacturer makes console and wide-screen televisions. The profit per unit is \(\$ 125\) for the console televisions and \(\$ 200\) for the wide-screen televisions. u. Let \(x=\) the number of consoles manufactured in a month and \(y=\) the number of wide-screens manufactured in a month. Write the objective function that describes the total monthly profit. b. The manufacturer is bound by the following constraints: \(\cdot\) Equipment in the factory allows for making at most 450 console televisions in one month. \(\cdot\) Equipment in the factory allows for making at most 200 wide-screen televisions in one month. \(\cdot\) The cost to the manufacturer per unit is \(\$ 600\) for the console telcvisions and \(\$ 900\) for the widescreen 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 non negative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at \((0,0),(0,200)\) \((300,200),(450,100), \text { and }(450,0) .]\) e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit by manufacturing ___console televisions each month ___ and \(_{-\infty}\) wide-screen televisions each month. The maximum monthly profit is ___.

In Exercises \(19-30,\) solve each system by the addition method. \(x+y=1\) \(x-y=3\)

Let \(x\) represent the first number, \(y\) the second number, and \(z\) the third number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The following is known about three numbers: Three times the first number plus the second number plus twice the third number is \(5 .\) If 3 times the second number is subtracted from the sum of the first number and 3 times the third number, the result is \(2 .\) If the third number is subtracted from 2 times the first number and 3 times the second number, the result is \(1 .\) Find the numbers.

In Exercises \(19-30,\) solve each system by the addition method. \(\begin{aligned} x+2 y &=2 \\\\-4 x+3 y &=25 \end{aligned}\)

You throw a ball straight up from a rooftop. The ball misses the rooftop on its way down and eventually strikes the ground. A mathematical model can be used to describe the relationship for the ball's height above the ground, \(y,\) after \(x\) seconds. Consider the following data: $$\begin{array}{|c|c|}\hline \begin{array}{c}x, \text { seconds after the } \\\\\text { ball is thrown }\end{array} & \begin{array}{c}y, \text { ball's height, in feet, } \\\\\text { above the ground }\end{array} \\ \hline 1 & 224 \\\\\hline 3 & 176 \\\\\hline 4 & 104 \\\\\hline\end{array}$$ a. Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. b. Use the function in part (a) to find the value for \(y\) when \(x=5 .\) Describe what this means.

In Exercises \(19-30,\) solve each system by the addition method. \(2 x+3 y=-16\) \(5 x-10 y=30\)

Use a graphing utility to sketch the region determined by the constraints. Then determine the maximum value of the objective function subject to the contraints. Objective Function \(\quad z=6 x+8 y\) Constraints \(\quad x \geq 0, y \geq 0\) \(x+2 y \leq 6\)

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(9 x-3 y=12\) \(y=3 x-4\)

A person invested \(\$ 17,000\) for one year, part at \(10 \%,\) part at \(12 \%,\) and the remainder at \(15 \% .\) The total annual income from these investments was \(\$ 2110 .\) The amount of money invested at \(12 \%\) was \(\$ 1000\) less than the amount invested at \(10 \%\) and \(15 \%\) combined. Find the amount invested at each rate.