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Chapter 8: Problem 107
If \(x=3, y=2,\) and \(z=-3,\) does the ordered triple \((x, y, z)\) satisfy the equation \(2 x-y+4 z=-8 ?\)
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For the linear function \(f(x)=m x+b, f(-2)=11\) and \(f(3)=-9 .\) Find \(m\) and \(b\).
Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. 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. \(\cdot\) 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.
Consider the objective function \(z=A x+B y(A>0\) 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 \(A=\frac{2}{3} B\).
Find the domain of each function. \(f(x)=\ln (6-x) \quad \text { (Section } 4.2, \text { Example } 10)\)
How many ounces of a 15% alcohol solution must be mixed with 4 ounces of a 20% alcohol solution to make a 17% alcohol solution?
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