Jeff tries to make his students happy. He passes out a questionnaire that lists a number of possible course policies in areas where he is flexible. Every student is asked to respond to each possible course policy with one of "strongly favor", "mostly neutral", or "strongly oppose". Each student may respond with "strongly favor" or "strongly oppose" to at most five questions. Because Jeff’s students are very understanding, each student is happy if (but only if) he or she prevails in at least one of their strong policy preferences. Either describe a polynomial-time algorithm for setting course policy to maximize the number of happy students, or show that the problem is NP-hard.

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5.a fishing rod is formed from a composite material of 0.5 kg of glass fibers embedded in a matrix of 0.5 kg of epoxy resin. the glass fibers are assumed to be long, continuous and unidirectional. to achieve a greater stiffness it is proposed to use a different composite that is comprised of long continuous carbon fibers that will be embedded in a matrix of 0.5 kg of epoxy resin. if the modulus of elasticity of the carbon fiber composite is 10% greater than the elastic modulus of the glass fiber composite, estimate the mass of carbon fibers that will be used to make the carbon fiber composite. assume the applied tensile stress is parallel to the direction of the long axis of the fibers. the epoxy resin, glass fiber, and carbon fiber have an elastic modulus of 5, 86, and 350 gpa respectively and a density of 1100, 2500, and 1800 respectively.

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