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MCV4U date: _______________________________ MAX/MIN ON AN INTERVAL ANSWERS: MCV4U1 MORE OPTIMIZATION date: _________________________ 1. Find the maximum and minimum value of each function on the given interval. a) f x b) f x ln x2 x 1, 2 x 2 ln x , 1 x e2 x d) f x c) f x x 2 e x , 0 x 2 3 8e x , ln 2 x ln 4 e 2x 4 t 20 2. The number of bacteria, N, in a culture at time t hours is N 200030 te . Find the largest number of bacteria in the culture during the interval 0 t 50 . 3. The net monthly profit from the sale of a certain product is given (in dollars) by the formula Px 106 1 x 1e 0.001x , where x is the number of items sold. a) Find the number of items that yield the maximum profit. Assume that at most 2000 items can be produced per month. b) Repeat part (a) assuming that at most 500 items can be produced per month. 4. The concentration of two medicines in the bloodstream t hours after injection are c1t tet and c2t t 2 e t . Which medicine has the larger maximum concentration? 5. The proportion of people who have responded to the advertisement of a new product after it has been marketed for t days is found to be 0.71 e0.2t . The marketing area contains 10 million potential customers, and each response to the advertisement results in a profit to the company of $0.70 (on average), excluding the cost of advertising. The advertising costs $30 000 to produce and $5000 per day to run. 0.2t a) Find lim 0.71 e and interpret the result. t b) c) d) e) What percentage of potential customers have responded after 7 days of advertising? Write the function Pt that represents the net profit after t days of advertising. What is the net profit after 28 days? For how many days should the advertising campaign be run in order to maximize profits? Assume an advertising budget of $180 000. What is the maximum profit? ANSWERS: 1 27 16 3 , min = 0 b) max = ln7, min = ln c) max = , min = 8 d) max = 2, min = e e 17 4 2. 74 715 at t = 20 hours 3.a) 1001 b) 500 4. the 2nd 5.a) 70% b) 52.7% c) Pt 10 7 0.7 2 1 e 0.2t 5000t 30000 d) $4 711 880.47 e) 26 days, $4 712 968 1.a) max = MCV4U1 OPTIMIZATION 1. Find two natural numbers whose sum is 16 and whose product is a maximum. 2. The sum of two natural numbers is 12. If the product of one number with the square of the other is a maximum find the numbers. 3. Find two positive numbers whose sum is 15 if the sum of their squares is a minimum. 4. The perimeter of a rectangle is 24 cm. Find the dimensions of the rectangle of maximum area. What is the maximum area? 5. The area of a rectangle is 64 cm 2 . Find the dimensions of the rectangle of minimum perimeter. What is the minimum perimeter? No interval is required for this solution. 6. Three sides of a rectangular field are to be fenced with 420 m of fencing. Find the dimensions of the field of largest area if the single fenced side must be at least as long as the two fenced sides. 7. A rectangular field is to be enclosed by a fence then subdivided into two areas by a fence parallel to the shorter side. If 600 m of fencing material is available and each side must be at least 75 m in length, find the largest possible total area that can be enclosed. 8. A rectangular field is to be enclosed by a fence then subdivided into three areas by fences parallel to the shorter side. Find the dimensions of the largest total area that can be enclosed with 800 m of fencing. 9. A rectangular box is made from a piece of cardboard which measures 48 cm by 18 cm by cutting equal squares from each corner and turning up the sides. Find the maximum volume of such a box if: a) the height of the box must be at most 3 cm. b) the length and width of the base must be at least 10 cm. 10. A piece of paper for a poster has an area of 1 m 2 . The margins at the top and bottom are 8 cm and at the sides are 6 cm. What are the dimensions of the sheet of paper which will maximize the printed area of the page? 11. An open topped box has a square base and vertical sides. If the surface area of the box is 108 m 2 find the dimensions that will maximize the volume if the side of the base must be at least 4 m long. 12. A fence is built around a rectangular lot and the lot then subdivided into two lots by a fence parallel to one of the sides. If the exterior fence costs $2.50/m and the interior fence costs $1.00/m, find the dimensions of the lot of maximum area that can be fenced for a total cost of $480. 13. A closed cylindrical can is constructed from a fixed amount of material. Determine the ratio of height to radius of the can with the maximum volume. 14. The volume of a closed cylindrical can is 25 m3 . Find the dimensions of the can with the minimum surface area. No interval is required for this solution 15. The cost per cm2 of the top and bottom of a closed cylinder is three times the cost per cm2 of the sides. Determine the ratio of the most economical dimensions of the container with volume 12 cm 3 . No interval is required for this solution. 16. The volume of a square based rectangular box is 252 dm3 . The construction cost of the bottom is $5.00 per dm2 , of the top is $2.00 per dm2 and of the sides is $3.00 per dm2 . Find the dimensions that will minimize the cost if the side of the base must fall between 4 dm and 8 dm. 17. Find the dimensions of the rectangle of maximum perimeter that can be inscribed in a circle of radius 4 cm. 18. Find the dimensions of the rectangle of maximum area that can be inscribed in an isosceles triangle with base 40 cm and height 30 cm. 19. Find the dimensions of the cylinder of maximum volume that can be inscribed in a cone with a diameter of 40 cm and a height of 30 cm. 20. Find the height and radius of the cylinder of greatest volume that can be inscribed in a sphere of radius R units. 21. A telephone company wants to run a cable from a point A on one bank of a river to a point B on the opposite bank and 12 km down stream from point A. The river is 5 km wide. The company can run the cable along the shore at a cost of $1 000 per km and across the river at a cost of $2 000 per km. What lengths of cable should be run along the shore and under the water to minimize the cost of the cable? 22. A power house cable runs from point P to a factory at point F located on the opposite bank of a river 200 metres wide and located 400 metres down stream from P. If it costs $12 per metre to lay the cable under the water and $6 per metre to run the cable along the bank find the length of the cable which runs under water if the total cost is a minimum. 23. At 10:00A.M., a ship is located 40 nautical miles due west of a second ship. If the first ship sails east at 20 knots and the second ship sails north at 10 knots when in the next two hours will the two ships be closest together? Find the minimum distance between the two ships. 24. A sailing ship is located 25 km due south of a drifting vessel. If the ship sails north at 4 km/h and the vessel drifts east at 3 km/h find the shortest distance between the vessels. 25. A wire of length 40 cm is cut into two pieces and bent to form a square and a circle. What two lengths will minimize the total area of the two figures? 26. A rectangle is inscribed in the ellipse 9 x 2 16 y 2 3600 with its sides parallel to the coordinate axes. Find the dimensions of the rectangle with the maximum perimeter. 27. A farmer estimates that if he digs his potato crop now he will have 120 bushels which he will be able to sell for $1 per bushel. If he waits, the crop will grow by 20 bushels per week while the price will drop by $0.10 per bushel per week. When should he dig his crop to maximize his return? 28. A telephone company finds that there is a net profit of $15 per phone if an exchange has 1000 subscribers. If there are more than 1000 their profits decrease by one cent for each subscriber over that number. How many subscribers will maximize profit. 29. A school trip will cost each student $15 if 150 students participate. However, the cost per student will be reduced by $0.05 for each student in excess of the 150. How many students should make the trip to maximize total income? 30. Find the point on the graph of y x which is closest to the point (1,0). 31. Find the minimum distance from the origin to the parabola y 5 x 2 . ANSWERS: 1. 8,8 36 cm2 2. 8,4 3. 7.5,7.5 4. 6 cm by 6 cm, 5. 8 cm by 8 cm, 32 cm 6. 210 m by 105 m 7. 15 000 m 2 8. 200 m by 100 m 9a) 1 512 cm3 9b) 1 600 cm3 11. 6 m by 6 m by 3 m 12. 48 m by 40 m 10. 2 3 3 m by m 3 2 3 25 100 m or m h 3 100 m 2 2 13. h:r=2:1 14. r 15. h:r=6:1 16. 6 dm by 6 dm by 7 dm 18. 15 cm by 20 cm 19. h 10 cm , r 21. 3 40 cm 3 10 3 36 5 3 km across river and km along shore. 3 3 23. 11:36AM , 8 5 n. mi. 24. 15 km 26. 32 units by 18 units 27. 2 weeks 29. 225 students 1 2 30. , 2 2 17. 4 2 cm by 4 2 cm 20. h 2 3R ,r 3 400 3 m 3 160 40 cm , cm 25. 4 4 22. 28. 1 250 subscribers 31. 19 units 2 6R 3