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Chapter 5: Problem 57
Find the length and width of a rectangle whose perimeter is 36 feet and whose area is 77 square feet.
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Consider the objective function \(z-A x+B y \quad(A>0\) and \(B>0\) ) subject to the following constraints: \(2 x+3 y \leq 9, x-y \leq 2, x \geq 0,\) and \(y \geq 0 .\) Prove that the objective function will have the same maximum value at the vertices \((3,1)\) and \((0,3)\) if \(A-\frac{2}{3} B\).
Involve supply and demand. Although Social Security is a problem, some projections indicate that there's a much bigger time bomb ticking in the federal budget, and that's Medicare. In \(2000,\) the cost of Social Security was \(5.48 \%\) of the gross domestic product, increasing by \(0.04 \%\) of the GDP per year. In \(2000,\) the cost of Medicare was \(1.84 \%\) of the gross domestic product, increasing by \(0.17 \%\) of the GDP per year. a. Write a function that models the cost of Social Security as a percentage of the GDP \(x\) years after 2000 . b. Write a function that models the cost of Medicare as a percentage of the GDP \(x\) years after 2000 . c. In which year will the cost of Medicare and Social Security be the same? For that year, what will be the cost of each program as a percentage of the GDP? Which program will have the greater cost after that year?
Group members should choose a particular field of interest. Research how linear programming is used to solve problems in that field. If possible, investigate the solution of a specific practical problem. Present a report on your findings, including the contributions of George Dantzig. Narendra Karmarkar, and L. G. Khachion to linear programming.
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: \(\cdot\) 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 \(\cdot\) 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.
Use the two steps for solving a linear programming problem, given in the box on page 577 , to solve the problems in Exercises 17鈥�23. A large institution is preparing lunch menus containing foods A and B. The specifications for the two foods are given in the following table: $$\begin{array}{cccc}\hline & \text { Units of Fat } & \text { Units of } & \text { Units of } \\\\\text { Food } & \text { per Ounce } & \text { Carbohydrates } & \text { Protein } \\\\\hline \mathrm{A} & 1 & \text { per Ounce } & \text { per Ounce } \\\\\mathrm{B} & 1 & 1 & 1 \\\\\hline\end{array}$$ Each lunch must provide at least 6 units of fat per serving, no more than 7 units of protein, and at least 10 units of carbohydrates. The institution can purchase food A for \(\$ 0.12\) per ounce and food \(\mathrm{B}\) for \(\$ 0.08\) per ounce. How many ounces of each food should a serving contain to meet the dietary requirements at the least cost?
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