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# ps1sln - Mathematics 116 Problem Set 1 Solutions Friday 28...

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Mathematics 116. Problem Set 1 Solutions Friday, 28 September 2007 1. We require x, y 0. The ingredient restrictions are 3 x + y 15 (chocolate bars) x + y 7 (pounds sugar) x + 2 y 12 (pounds flour) These restriction define a convex set with vertices at A = (0 , 6), B = (2 , 5), C = (4 , 3), D = (5 , 0), and E = (0 , 0). The line AB has slope - 1 2 (the flour restriction), the line BC has slope - 1 (the sugar restriction), and the line CD has slope - 3 (the chocolate restriction). To maximize profit, we find the highest c for which the line ax + by = c touches the feasible set. (a) The line 6 x + y = c has slope - 6, so it first touches the set at x = 5, y = 0. (b) The line 3 x + 2 y = c has slope - 3 2 , so it first touches the set at (4 , 3). (c) The line 2 x + 3 y = c has slope - 2 3 , so it first touches the set at (2 , 5). (d) The line x + 5 y = c has slope - 1 5 , so it first touches the set at (0 , 6). (e) The line 2 x + 2 y = c has slope - 1, so it touches the set along the line BC . To maximize profit, the only requirement is that we use up all the sugar. 2. (a) We want to maximize the function - x 2 - y 2 + 8( x - 5) under the constraints x 5, x + y 10. Ignoring the constraints, the only critical point of the function is (4 , 0) which does not satisfy the constraints, so we know that we are looking for a boundary solution.

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ps1sln - Mathematics 116 Problem Set 1 Solutions Friday 28...

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