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Applications of Extrema

Applications of Extrema. Lesson 6.2. A Rancher Problem. You have 500 feet of fencing for a corral What is the best configuration (dimensions) for a rectangular corral to get the most area One side of the rectangle already has a fence. Sample Problem.

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Applications of Extrema

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  1. Applications of Extrema Lesson 6.2

  2. A Rancher Problem • You have 500 feet of fencing for a corral • What is the best configuration(dimensions) for a rectangularcorral to get the most area • One side of the rectangle already has a fence

  3. Sample Problem • Your assistant presents you witha contract for signature • Your firm offers to deliver 300 tables to a dealer at $90 per table and to reduce the price per table on the entire order by $0.25 for each additional table over 300 • What should you do? • Find the dollar total involved in largest (smallest) possible transaction between the manufacturer and the dealer.

  4. Solution Strategy • Read the problem carefully • Make sure you understand what is given • Make sure you see what the unknowns are • From our problem • Given • 300 tables at $90 per table • $0.25 reduction per table on entire order if > 300 • Unknowns • Largest possible transaction • Smallest possible transaction

  5. Solution Strategy • If possible sketch a diagram • Label the parts • From our problem • Not much to diagram … • More likely in a problem about the size of a box to minimize/maximize materials or volume x + 3 x 2x

  6. Solution Strategy • Decide on a variable to be maximized (minimized) • Express variable as a function of one other variable • Be sure to find function domain • From our problem • T = transaction amount • T = f(x) = ?

  7. Solution Strategy • To analyze the function, place it in Y= screen of calculator • Check the table (♦Y) to evaluate the domain and range for setting the graph window

  8. Solution Strategy • Find the critical points for the function • View on calculator • For our problem • Use derivative tests to find actual points Note jump in discontinuous function

  9. Solution Strategy • If domain is closed interval • Evaluate at endpoints, critical points • See which value yields absolute max or min • For our problem maximum minimum

  10. Strategy Review • Read carefully, find knowns, unknowns • Sketch and label diagram • Determine variable to be max/min • Express as function of other variable • Determine domain • Find critical points • If domain is closed interval • Check endpoints • Check critical points

  11. Practice Problem • A fence must be built to enclose a rectangular areaof 20,000 ft2 • Fencing material costs $3/ft for the two sides facing north and south • It costs $6/ft for the other two sides • Find the cost of the least expensive fence

  12. Assignment • Lesson 6.2 • Page 383 • Exercises 5 – 33 odd

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