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Optimization Problems in Economics and Geometry

The content discusses optimization problems in economics, focusing on maximizing profit by analyzing revenue and cost functions. It also presents geometric optimization problems such as finding optimal travel routes, maximizing area in triangles, minimizing fold length on paper, and determining the least area cut off by a line. Various strategies and equations are outlined to address these optimization challenges.

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Optimization Problems in Economics and Geometry

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  1. MATH 3 Calculus Scott Pauls Department of Mathematics Dartmouth College

  2. 1. Root One: Jasmin, Kyerewaa, Lee, Tan, Morse. Braelyn 2. 2∞& beyond: Braceras, Donahue, Fisli, Laverty, Swett. Braelyn Braelyn 3. The functions: Krass, Ram Sontag, Sullivan, Vitcov. Zach 1 5 4 2 4. Live free or ?: Archer, Broom, Mitchell, Okonta, Unger, Anderson. Braelyn 5. Good vibes only: Garcia, Kalk, Khan, Nathoo, Sapone. Braelyn 11 12 8 6 Zach Serena 6. INTEGR8: Batista, Christie, Redmond, Tronsson, Urquiza. Zach 9 7 10 3 7. Lone Pine: Dorsey, Ingabire, Jung, Kovary, Strong. Serena Front of the classroom 8. Limits out for Harambe: Kanagala, Kusnarowis, Singh, Sofio, Straus. Serena 9. PEMDAS: Eng, Han, Miller, Poomkudy, Zaun, Press. Zach 10. Union Jacks: Abass, Fear, John, McMillian, Potter. Serena 11. Apple ?: Almonte, Arifi, Greaney, Johnson, Moore. Serena 12. #FreeKodak: Correa, Cullen, Douglas, Higgins, Tuthill. Zach

  3. Optimization: Economics In basic economic analysis, we state profit in terms of revenue and cost: ? ? = ? ? − ?(?) where ? is the number of units sold. Revenue is given in terms of the price of one unit of the good: ? ? = ??(?) where ?(?) is the price of ? units (?(?) is also called the demand function). 1. Write an outline for finding the number of units what maximize the profit. 2. For what types of price and cost functions do you think you’d be able to execute the outline? 3. If ? ? = 1700 − 7? and ? ? = 16000 + 500? − 1.6?2+ 0.004?3, what value of ? maximizes the profit?

  4. Optimization problems 1. A woman stands on the edge of a circular lake with a radius of 2 miles. She can walk along the edge at a rate of 4 miles/hour and row a boat across the lake at a rate of 2 miles/hour. What route should she take to minimize the time to travel to the point directly across from her? 2. Show that of all the isosceles triangles with a given perimeter, the one with the greatest area is equilateral. 3. The upper right-hand corner of a piece of paper, 12 in. by 8 in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose to minimize? 4. Find an equation of the line through the point (3,5) that cuts off the least area from the first quadrant.

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