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Optimization Note Examples
1. Farmer McDonald has 800 m of fencing and wishes to enclose a rectangular field. One side of the field is against a river and does not need fencing. Find the dimensions of the field if the fenced area is to be a maximum. A county fair has a holding area for prize sheep that are entered in a contest. The holding area is made up of 12 identical pens arranged in a two by six grid. If 100 m of fencing is available, what dimensions of each pen would maximize the total holding area? State EXACT answer. (from McGraw-Hill Ryerson Calculus & Advanced Functions Pg. 382 #12b) A piece of sheet metal 60 cm by 30 cm is to be used to make a rectangular box with an open top. Squares are to be cut from each corner of the sheet metal, the sides folded upward to form the box, and then the seams welded. Find the dimensions that will give the box with the largest volume. State EXACT answer. An enclosed can is to have a capacity of 1000 cm 3 . Find the dimensions of the can if a minimum amount of tin is to be used. The marketing department has decided that: a) its height must be at least 4 cm. b) its height must be at least 4 cm and its diameter must be at least 6 cm. Answer accurate to one decimal place. 5. An amount of $1000 is invested at 5% compounded annually. a) Write an equation for the amount of the investment, A, for t years. b) Determine the rate of change in the amount of the investment with respect to time, t. c) If the money is invested for 8 years, i) At what time is the amount a minimum? What is the minimum amount? ii) At what time is the amount a maximum? What is the maximum amount? The effectiveness of studying for an exam depends on how many hours a student studies. Some experiments show that if the effectiveness, E, is put on a scale of 0 to 10, then −t E ( t ) = 0.5 10 + te 20 , where t is the number of hours spent studying for an examination. If a student has up to 30 hours for studying, how many hours are needed for
1. Farmer McDonald has 800 m of fencing and wishes to enclose a rectangular field. One side of the field is against a river and does not need fencing. Find the dimensions of the field if the fenced area is to be a maximum. A county fair has a holding area for prize sheep that are entered in a contest. The holding area is made up of 12 identical pens arranged in a two by six grid. If 100 m of fencing is available, what dimensions of each pen would maximize the total holding area? State EXACT answer. (from McGraw-Hill Ryerson Calculus & Advanced Functions Pg. 382 #12b) A piece of sheet metal 60 cm by 30 cm is to be used to make a rectangular box with an open top. Squares are to be cut from each corner of the sheet metal, the sides folded upward to form the box, and then the seams welded. Find the dimensions that will give the box with the largest volume. State EXACT answer. An enclosed can is to have a capacity of 1000 cm 3 . Find the dimensions of the can if a minimum amount of tin is to be used. The marketing department has decided that: a) its height must be at least 4 cm. b) its height must be at least 4 cm and its diameter must be at least 6 cm. Answer accurate to one decimal place. 5. An amount of $1000 is invested at 5% compounded annually. a) Write an equation for the amount of the investment, A, for t years. b) Determine the rate of change in the amount of the investment with respect to time, t. c) If the money is invested for 8 years, i) At what time is the amount a minimum? What is the minimum amount? ii) At what time is the amount a maximum? What is the maximum amount? The effectiveness of studying for an exam depends on how many hours a student studies. Some experiments show that if the effectiveness, E, is put on a scale of 0 to 10, then −t E ( t ) = 0.5 10 + te 20 , where t is the number of hours spent studying for an examination. If a student has up to 30 hours for studying, how many hours are needed for