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Chapter 8: Problem 4
Graph each inequality. $$2 x-y>4$$
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$$\text { If } f(x)=5 x^{2}-6 x+1, \text { find } \frac{f(x+h)-f(x)}{h}$$ (Section 2.2, \text { Example } 8)
You invest in a new play. The cost includes an overhead of \(\$ 30,000,\) plus production costs of \(\$ 2500\) per performance. A sold-out performance brings in \(\$ 3125 .\) (In solving this exercise, let \(x\) represent the number of sold-out performances.
will help you prepare for the material covered in the next section. Solve by the addition method: $$\left\\{\begin{array}{l}{2 x+4 y=-4} \\\\{3 x+5 y=-3}\end{array}\right.$$
Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. In 1978 , a ruling by the Civil Aeronautics Board allowed Federal Express to purchase larger aircraft. Federal Express's options included 20 Boeing 727 s that United Airlines was retiring and/or the French-built Dassault Fanjet Falcon \(20 .\) To aid in their decision, executives at Federal Express analyzed the following data: $$\begin{array}{ll} {} & {\text { Boeing } 727 \quad \text { Falcon } 20} \\ {\text { Direct Operating cost }} & {\$ 1400 \text { per hour } \$ 500 \text { per hour }} \\ {\text { Payload }} & {42,000 \text { pounds } \quad 6000 \text { pounds }} \end{array}$$ Federal Express was faced with the following constraints: \(\cdot\) Hourly operating cost was limited to 35,000. \(\cdot\) Total payload had to be at least 672,000 pounds. \(\cdot\) Only twenty 727 s were available. Given the constraints, how many of each kind of aircraft should Federal Express have purchased to maximize the number of aircraft?
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 Constraints $$ \begin{aligned} &z=5 x+6 y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {2 x+y \geq 10} \\ {x+2 y \geq 10} \\ {x+y \leq 10} \end{array}\right. \end{aligned} $$
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