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Optimization

Optimization. Many times in life we are asked to do an optimization problem – that is, find the largest or smallest value of some quantity…. Find the route which will minimize the time it takes me to get to school Build a structure with the least amount of material

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Optimization

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  1. Optimization

  2. Many times in life we are asked to do an optimization problem – that is, find the largest or smallest value of some quantity… • Find the route which will minimize the time it takes me to get to school • Build a structure with the least amount of material • Build a structure costing the least amount of money • Find the least medication one should take to help with a medical problem. • Find the most a company should charge for a CD in order to make as much money as possible. • Build a yard enclosure with the most amount of space.

  3. A company that manufactures bicycles estimates that the profit for selling a particular model is P(t)= -45x3 + 2500x2- 275,000 where P is the profit (in dollars) and x is the advertising expense (in tens of thousands of dollars). According to this model, how much money should they spend on advertising to maximize their profit?

  4. If a farmer has 250 feet of fence and wants to make a rectangular pen with a divider to separate animals, one side of which is along the side of his barn, what dimensions should be used in order to maximize the area of the pen?

  5. The Box Competition • Create an open top box by cutting congruent squares from the corners of a piece of paper measuring 22.8cm by 30.5cm. • CHALLENGE… The group whose box has the largest volume WINS!

  6. The Box Competition Rules • You may use only the supplies provided for you by Mrs. Chumas • You will have 15 minutes to create your box and find its volume. • Be sure to record the side length of your congruent squares, the dimensions of your box and the volume of your box on your whiteboard.

  7. Box Problem • What are the dimensions of the box containing the largest volume that can be made by cutting congruent squares from the corners of a piece of paper measuring 22.8cm by 30.5cm ? (Assume the box has no lid)

  8. Summary of the steps… Step 1: Name the variables (there are usually 2) and write down any relationship between the variables (this is the secondary equation). Step 2: Write down the function that is to be maximized or minimized, giving it a name (this is the primary equation). Step 3: Reduce the number of variables in the function you are maximizing or minimizing (primary equation) to one variable using the relation written in step 1. Completely simplify this equation! Step 4: Find the critical number(s) of your function and verify whether they give max or min using the first derivative number line and answer the question.

  9. A soccer field will be fenced in at McAlpine Park in the near future. No fence will be required on the side lying along the creek. If the new wood fence costs $12 per meter for the side parallel to the creek, and $4 per meter for the other two sides, find the dimensions of the soccer field of maximum area that could be enclosed with a budget of $3,600.

  10. If a closed can with volume 24π cubic inches is made in the form of a cylinder, find the height and radius of the can with minimum surface area.(V= πr2h, A= 2πr2 + 2πrh)

  11. A rectangular box is to be made with a square base and having a volume of 100 in3. If the cost of the materials are $1 per square inch for the bottom, $1.50 per square inch for the sides and $2 per square inch for the top, what dimensions will yield the least expensive box?

  12. Rowboat Problem • You are in a rowboat on Lake Erie, 2 miles from a straight shoreline, taking your potential in-laws for a boat ride. Six miles down the shoreline from the nearest point on the shore is an outhouse. You suddenly feel the need for its use. It is October, so the water is too cold to go in, and besides, your in-laws are already pretty unimpressed with your “yacht”. It wouldn’t help matters to jump over the side to relieve your distended bladder. Also, the shoreline is populated with lots of houses, all owned by people who know your parents, and would love to get you in trouble with them. If you can row at 2 mph and run at 6 mph (you can run faster when you don’t have to keep your knees together), for what point along the shoreline should you aim in order to minimize the amount of time it will take you to get to the outhouse?

  13. Rowboat Problem • Henry, who is in a rowboat 2 miles from the nearest point B on a straight shoreline, notices smoke billowing from his house, which is 6 miles down the shoreline from B. He figures he can row at 6mph and run 10mph. How should he proceed in order to get to his house in the least amount of time? 6 miles B 2 miles

  14. Exploiting Humanity ProblemBONUS QUESTION • You have just invented a new peanut butter guacamole dip, and you open a stand in front of the student union to sell this goop by the jar. Somehow a rumor gets started, certainly not traceable back to you, that it is an aphrodisiac, and sales take off. At a price of $1 a jar, you sell 500 jars a day. For each nickel that you increase the price, you sell 2 fewer jars. Assuming that your fixed cost per day is $200 (protection money), and the cost per jar to you is 50 cents, determine the price for which you should sell your dip in order to maximize your profit.

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