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Chapter 5: Problem 10
Sketch the graph of the inequality. $$y^{2}-x<0$$
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Reasoning An objective function has a minimum value at the vertex \((20,0)\). Can you conclude that it also has a minimum value at the point \((0,0)\) ? Explain.
Reasoning An objective function has a maximum value at the vertices \((0,14)\) and \((3,8)\). (a) Can you conclude that it also has a maximum value at the point \((1,12)\) ? Explain. (b) Can you conclude that it also has a maximum value at the point \((4,6)\) ? Explain. (c) Find another point that maximizes the objective function.
Sketch the graph of the inequality. $$x \geq 2$$
Optimal Profit A company makes two models of a patio furniture set. The times for assembling, finishing, and packaging model \(\mathrm{A}\) are 3 hours, \(2.5\) hours, and \(0.6\) hour, respectively. The times for model \(\mathrm{B}\) are \(2.75\) hours, 1 hour, and \(1.25\) hours. The total times available for assembling, finishing, and packaging are 3000 hours, 2400 hours, and 1200 hours, respectively. The profit per unit for model \(\mathrm{A}\) is $$\$ 100$$ and the profit per unit for model \(\mathrm{B}\) is $$\$ 85 .$$ What is the optimal production level for each model? What is the optimal profit?
Find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. (For each exercise, the graph of the region determined by the constraints is provided.) Objective function: $$ z=2 x+8 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 2 x+y & \leq 4 \end{aligned} $$
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