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4. Consider a glass factory that can produce four different kinds of glasses: beer, wine, drinking, and schnapps. Glass manufacturing uses machine hours, labour, and sand as its main inputs. The demand for each type of glass depends on the price set by the factory. To make matters simple, assume that demand is as follows: Beer up to 700 units at price ≤ 500, and nothing for price > 500 Wine up to 1100 units at price ≤ 500, and nothing for price > 500 Drinking up to 900 units at price ≤ 350, and nothing for price > 350 Schnapps up to 1400 units at price ≤ 320, and nothing for price > 320 Hence, the factory would simply charge 500 for a beer glass, 500 for wine, 350 for drinking and 320 for Schnapps. Production technology is described by the following table: Machine Hours (hrs/unit) Labour Hours (hrs/unit) Sand (ton/unit) Beer 1.0 10 0.5 Wine 0.7 12 0.4 Drinking 0.8 6 0.5 Schnapps 0.5 7 0.3 We consider two scenarios. In the first scenario (A), labour and sand are available in unlimited supply at the prices of $20/hr and $100/ton, while machine hours are in fixed supply: a maximum of 2,000 hours. In the second scenario (B), also labour hours are in fixed supply: a maximum of 25,000 hours. a) Set up a linear program for each of the two scenarios (A and B) to determine the optimal product mix for the glass factory. b) Explain what the shadow prices mean in each case. In scenario A, explain what the reduced cost for drinking glasses mean. c) In scenario B, find out how high (at least) the factory should set the price of a champagne glass that uses inputs as follows: 1.2 hours of machine time, 11 hours of labour time and 0.4 tons of sand per unit of production. Hint: Add champagne glass as an additional product that the factory can produce and initially charge a price of zero for it. What does the reduced cost for champagne glass mean?
Determine the probability distribution, and the cumulative probability distribution of car arrivals. Simulate 20 hours of car arrivals at Joe Kelly's oil change and tune-up place.
Suppose that an object is projected in such a way that its height above the ground in feet after t seconds is given by s(t) = -16t2 +48t + 46. Find the maximum height of the object.
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a moving truck rental company has a total of 24 trucks in its lot. The trucks are either 4-wheelers or 18-wheelers. How many of each kind of truck does the company have if the total number of wheels is 138?
How can you represent a pedigree for a mixed breed of dog (a Schnoodle, for example) using set theory? Create (describe) a Venn diagram and a numeric sentence to describe the set.
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Let R be a commutative ring with 1. Prove that if (a,b)=1 and a divides bc, then a divides c. More generally, show that if a divides bc with nonzero a,b then a/(a,b) divides c.
Circle O and circle P are externally tangent circles. Circle O is tangent to circle P at T, OT = 12, TP = 6. AB is a common external tangent line to both circles. Find AB.
If a person stands at the end of the shadow and looks up to the top of the building, what is the angle of the person's eyes to the top of the building (to the nearest hundredth of a degree)? (Assume the person's eyes are 5 feet above ground level...
A searchlight is shaped like a paraboloid of revolution. If the light source is located 5 feet from the base along the axis of symmetry and the opening is 8 feet across, how deep should the searchlight be?
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