MTH Calculus with Analytic Geom I Project 1

Transcription

1 MTH Calculus with Analytic Geom I Project 1 Project Instructions: 1. Mathematics: Appropriate calculus skills and mathematics must be applied. 2. Representation: Thorough Analytical and graphical explanations of problem. 3. Construction: Constructions must be done with precise measurements. 4. Neatness: All work must be legible and easy to understand. Plagiarism is strictly prohibited in this project work. You are neither allowed to consult anyone ( except group members, or your instructor) nor use any outside material except your textbook. Your final report must contain all software and other sources used. Final Submission and Presentation: Final report of project work must be TYPED using MS-WORD, LA- TEX or any format and submitted online on BLACKBOARD. You must also create a presentation slide (using powerpoint or similar format) for your presentation. Instructions on how to submit on BLACKBOARD will be posted online. 1

2 Project 1:1 Pizza Hut has just introduced a larger personal size pizza. The Pizza Hut Company needs a design for a box to package this new pizza. The box is to be made from a piece of cardboard of size 10 inches by 18 inches. The company wants to design a box with the largest volume. Six squares are to be cut from the cardboard as shown in Figure 1. By using the application of calculus, find the dimension of the box with largest volume. The company will also like you to construct a prototype of the box with such dimensions using a piece of cardboard Figure 1: 2

3 Project 1:2 A norman window has the outline of a semicircle on top of a rectangle as shown in Figure 2. Suppose there is 9 + π feet of wood trim available for all 4 sides of the rectangle and the semicircle, find the dimensions of the rectangle and semicircle that will maximize the area of the window. Figure 2: 3

4 Project 1:3 A rectangular sheet of paper with perimeter 36 cm is to be rolled into a cylinder. What are the dimensions of the sheet that give the greatest volume? 4

5 The corn yield on a certain farm is Project 1:4 Y = 0.118x x bushels per acre where x is the number of corn plants per acre (in thousands). Assume that corn seed costs $1.25 (per thousand seeds) and that corn can be sold for $1.50/bushel. Let P (x) be the profit (revenue minus the cost of seeds) at planting level x. a). Compute P (x 0 ) for the value x 0 that maximizes yiels Y. b). Find the maximum value of P (x). c). Does maximum yield lead to maximum profit? 5

6 Project 1:5 Find the maximum area of the rectangle inscribed in the region bounded by the graph of y = 4 x 2+x and the axes. (see Figure below). 6

7 MTH Calculus with Analytic Geom I Project 2 Project Instructions: 1. Mathematics: Appropriate calculus skills and mathematics must be applied. 2. Representation: Thorough Analytical and graphical explanations of problem. 3. Construction: Constructions must be done with precise measurements. 4. Neatness: All work must be legible and easy to understand. Plagiarism is strictly prohibited in this project work. You are neither allowed to consult anyone ( except group members, or your instructor) nor use any outside material except your textbook. Your final report must contain all software and other sources used. Final Submission and Presentation: Final report of project work must be TYPED using MS-WORD, LA- TEX or any format and submitted online on BLACKBOARD. You must also create a presentation slide (using powerpoint or similar format) for your presentation. Instructions on how to submit on BLACKBOARD will be posted online. 1

8 Project 2:1 Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than over land because air generally rises over land and falls over water during the day/ A bird with these tendencies is released from an island that is 5km from the nearest point B on a straight shoreline, flies to a point C on the shoreline, and then flies along the shoreline to its nesting area D. Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points B and D are 13 km apart. a). In general, f it takes 1.4times as much energy to fly over water as it does over land, to what point C should the bird fly in order to minimize the total energy expenditure in returning to its nesting area? b). Let W and L denotes the energy (in joules) per kilometer flown over water and land, respectively. Determine the ratio W/L corresponding to the minimum expenditure energy. c). What should W/L be in order for the bird to fly i). directly to its nesting area D? ii). to B and then along the shoreline to D? d). If the ornithologists observe that the birds of a certain species reach the shore at a point 4 km from B, how many times more energy does it take a bird to fly over water than over land? Figure 1: 2

9 Project 2:2 A company plans to introduce a new right circular cylindrical open-top cup for their soft drink. The cup is to hold 266 cubic centimeter of liquid when filled to the top. The company wants your team to design a cup that can be constructed with the least amount of material so that they can cut back on production cost. What are the dimensions of the cup? The company will also like you to construct a prototype of the cup with such dimensions using a piece of cardboard.construct the cup using a piece of cardboard. 3

10 Project 2:3 Your group is assigned to design a rectangular industrial warehouse consisting of three separate spaces of equal size as shown in Figure 2. The wall materials cost $500 per meter and a company allocates 2, 400, 000 for this project. a). Which dimension maximize the area of the warehouse? b). What is the area of each compartment in this case? Figure 2: 4

11 Project 2:4 In 1535 the mathematician Antonio Fior challenged his rival Nicolo Tartaglia to solve this problem: A tree stands 12 braccia high; it is broken into two parts at such a point that the height of the part left standing is the cube root of the length of the part cut away. a). What is the height of the part left standing? b). Show that this is equivalent to solving x 3 + x = 12. c). Find the exact height x of the part left standing to nine decimal places. (state the method used). Show that your answer is equivalent to the exact solution given by x = 3. 9 d). Find the exact height x of the part cut away to nine decimal places. (state the method used). 5

12 Project 2:5 A four-wheel-drive vehicle is transporting an injured hiker to the hospital from a point that is 30km from the nearest point on a straight road. The hospital is 50 km down the road from the nearest point. If the vehicle can drive at 30kph over the terrain and at 120kph on the road, how far down the road should the vehicle aim to reach the road to minimize the time it takes to reach the hospital? 6

13 MTH Calculus with Analytic Geom I Project 3 Project Instructions: 1. Mathematics: Appropriate calculus skills and mathematics must be applied. 2. Representation: Thorough Analytical and graphical explanations of problem. 3. Construction: Constructions must be done with precise measurements. 4. Neatness: All work must be legible and easy to understand. Plagiarism is strictly prohibited in this project work. You are neither allowed to consult anyone ( except group members, or your instructor) nor use any outside material except your textbook. Your final report must contain all software and other sources used. Final Submission and Presentation: Final report of project work must be TYPED using MS-WORD, LA- TEX or any format and submitted online on BLACKBOARD. You must also create a presentation slide (using powerpoint or similar format) for your presentation. Instructions on how to submit on BLACKBOARD will be posted online. 1

14 Project 3:1 Your group s job, as a consultant to a candy maker, is to design a jelly bean boxes each having a volume of 1200 inch 3. Each of these jelly bean boxes is to be an open-topped rectangular box with a square base having edge length x. In addition, the box is to have a square lid with a 2-inch rim. Thus the box-with-lid actually consists of two-opened boxes-the candy box itself with height y-inch, and the lid with height 2 in (we assume the lid fits very snugly). If it costs $9 per square inch each to construct the opentopped rectangular box and the square lid. The candy maker s problem is to determine the dimensions x and y that will minimize the total cost of production. Figure 1: The company will also like you to construct a prototype of the box with such dimensions using a piece of cardboard. 2

15 Project 3:2 Figure 2: 3

16 Project 3:3 A box with a square base and open top (shown below) must have a volume of 32, 000 cm 3. Find the dimensions of the box that minimize the amount of material used (i.e. surface area). Figure 3: 4

17 Project 3:4 A box of volume 8m 3 with a square top and bottom is constructed out of two types of metal. The metal for the top and bottom costs $50/m 2 and the metal for the sides costs $30/m 2. Find the dimensions of the box that minimize total cost. 5

18 Project 3:5 According to a model developed by economists E. Heady and J. Pesek, if fertilizer made from N pounds of nitrogen and P pounds of phosphate is used on an acre of farmland, then the yield Y of corn (in bushels per acre) is Y = N + 0.7P 0.001N P NP. A farmer intends to spend 30 per acre on fertilizer. If nitrogen costs 25 cents/lb and phosphate costs 20 cents/lb, your group is to come up with a combination of N and P that produces the highest yield of corn? 6

19 MTH Calculus with Analytic Geom I Project 4 Project Instructions: 1. Mathematics: Appropriate calculus skills and mathematics must be applied. 2. Representation: Thorough Analytical and graphical explanations of problem. 3. Construction: Constructions must be done with precise measurements. 4. Neatness: All work must be legible and easy to understand. Plagiarism is strictly prohibited in this project work. You are neither allowed to consult anyone ( except group members, or your instructor) nor use any outside material except your textbook. Your final report must contain all software and other sources used. Final Submission and Presentation: Final report of project work must be TYPED using MS-WORD, LA- TEX or any format and submitted online on BLACKBOARD. You must also create a presentation slide (using powerpoint or similar format) for your presentation. Instructions on how to submit on BLACKBOARD will be posted online. 1

20 Project 4:1 2

21 Project 4:2 The rectangular plot (Figure below) has size 100m 200m. Pipe is to be laid from A to a point P on side BC and from there to C. The cost of laying pipe along the side of the plot is $45 per meter and the cost through the plot is $80/m (since it is underground). a). Let f (x) be the total cost, where x is the distance from P to B. Determine f (x), but note that f is discontinuous at x = 0 (when x = 0, the cost of the entire pipe is $45 per feet) b). What is the most economical way to lay the pipe? What if the cost along the sides is $65 per meter? 3

22 Project 4:3 A swimmer launches his boat from point A on a bank of a straight river, 3km wide, and wants to reach point B, 8 km downstream on the opposite bank, as quickly as possible. He could proceed in any of three ways: 1. Row his boat directly across the river to point C and then run to point B 2. Row directly to point B 3. Row to some point D between C and B and then run to point B. If he can row 6km/h and run 8 km/h, where should he land to reach point B as soon as possible? Figure 1: 4

23 Project 4:4 A truck gets 10 miles per gallon (mpg) of diesel fuel traveling along an interstate highway at 50 mph. This mileage decreases by 0.15 mpg for each mile per hour increase above 50 mph. a). If the truck driver is paid $30/h and diesel fuel costs P = $3/gal, which spead v between 50 and 70 mph will minimize the cost of a trip along the highway? Notice that the actual cost depends on the length of the trip, but the optimal speed does not. b). Plot cost as a function of v (choose the length arbitrarily) and verify your answer to part (a). c). Do you expect the optimal speed v to increase or decrease if fuel costs go down to P = $2/gal? 5

24 Project 4:5 What is the maximum area of a rectangle inscribed in a right triangle (as shown below) with legs of length 3 and 4? The sides of the rectangle are parallel to the legs of the triangle. 6