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Chapter 4: Problem 46
Find a system of inequalities whose solution set is empty.
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f Purchasing Federal Rent-a-Car is putting together a new fleet. It is considering package offers from three car manufacturers. Fred Motors is offering 5 small cars, 5 medium cars, and 10 large cars for \(\$ 500,000\). Admiral Motors is offering 5 small, 10 medium, and 5 large cars for \(\$ 400,000\). Chrysalis is offering 10 small, 5 medium, and 5 large cars for \(\$ 300,000\). Federal would like to buy at least 550 small cars, at least 500 medium cars, and at least 550 large cars. How many packages should it buy from each car maker to keep the total cost as small as possible? What will be the total cost?
You are thinking of making your home more energy efficient by replacing some of the light bulbs with compact fluorescent bulbs, and insulating part or all of your exterior walls. Each compact fluorescent light bulb costs \(\$ 4\) and saves you an average of \(\$ 2\) per year in energy costs, and each square foot of wall insulation costs \(\$ 1\) and saves you an average of \(\$ 0.20\) per year in energy costs. \(^{12}\) Your home has 60 light fittings and 1,100 sq. ft. of uninsulated exterior wall. You can spend no more than \(\$ 1,200\) and would like to save as much per year in energy costs as possible. How many compact fluorescent light bulbs and how many square feet of insulation should you purchase? How much will you save in energy costs per year?
Can the value of the objective function decrease in passing from one tableau to the next? Explain.
$$ \begin{array}{ll} \text { Maximize } & p=x+5 y \\ \text { subject to } & x+y \leq 6 \\ & -x+3 y \leq 4 \\ & x \geq 0, y \geq 0 . \end{array} $$
In February 2008 , each episode of "American Idol" was typically seen by \(28.5\) million viewers, while each episode of "Back to You" was seen by \(12.3\) million viewers. \(^{15}\) Your marketing services firm has been hired to promote Bald No More's hair replacement process by buying at least 30 commercial spots during episodes of "American Idol" and "Back to You." The cable company running "American Idol" has quoted a price of \(\$ 3,000\) per spot, while the cable company showing "Back to You" has quoted a price of \(\$ 1,000\) per spot. Bald No More's advertising budget for TV commercials is \(\$ 120,000\), and it would like no more than \(50 \%\) of the total number of spots to appear on "Back to You." How many spots should you purchase on each show to reach the most viewers?
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