Assignment 3 - Optimization - I60 Instructions [Assessment 0 Write your complete solution below each question or on a separate sheet of paper. Number each question clearly a Show correct understanding of the problem and draw/label the diagram as appropriate (~2 marks). 0 Dene any variables and formulate equations of all relationships I functions correctly (~2 marks) Use proper notation (-2 marks) Follow the Optimization & Extreme Value Algorithm (p. 135, p. 144) {-2 marks) Show all steps clearly and provide proper reasoning as appropriate (-2 marks). Put a box around the nal answer for each question. Question 1 - Minimize Perimeter I 10 marks Athree-sided fence is to be built next to a straight section of the AternoPescara river, which forms the fourth side of a rectangular region, as shown in the diagram. The enclosed area is to equal 1300 m2 and the fence running parallel to the river must be set back at least 2|] m from the river. Determine the minimum perimeter of such an enclosure and the dimensions of the corresponding enclosure. Question 2 - Shortest Time I 10 marks Holokai is a small Island 5 km north of Birch Point, located along a straight shoreline. A triathlete, Melia, is camped at a spot on the shore that is 12 km west of Birch Point and she needs to go to the island. Melia can run at a rate of 8kmlh and can swim at a rate of 3kah. How far should Melia run (along the shoreline) before swimming to reach the island in the shortest amount of time? Question 3 - Minimize Cost I 10 marks A50 km stretch of highway runs west to east between two towns, Bamyan and Sumara. There are two new housing developments south of these two towns. Jamrin is located 30 km south of Bamyan and Zimrin is located 40km south of Sumara. The local government is planning to connect a road from each of the new settlements (Jamrin & Zimrin) to a common point on the existing highway. Find the point on the existing Jamrin highway to make the connection to minimize the total length of the new roads. Bamyan 3 U Question 4 - Motion of a Marble I 10 marks 3 2 _ The position function of a marble moving along a track is 5(t) = % (5 t ) t 2 for t S 10 a) Find the marble's velocity and acceleration function. b) When does the marble return to its starting position, 5(0)? c) At what time(s) is the marble at rest? d) During what intervals are s(t), v(t), and a(t) positive and when are they negave? Summarize it in a chart (similarto what is shown in the next question). Question 5 - Sketch a Graph I 10 marks a) Sketch (plot) the graphs of position, velocity, and acceleration of a particle (for 0 S t S 8 )that has the following characteristics. 1 0 2 4 6 b) Come up with (the equation of) a position function that meets the above criteria. Explain how it meets all of the above conditions. Question 6 - Maximize Profit I 10 marks Please review the attached note before solving the problem. A store sells 2000 action gures a month at a price of $15 each. After conducting market research, the company believes that sales will increase by 200 for each $0.20 decrease in price. a) Determine the demand function d(x). (To avoid confusion let's call our demand function d(x) instead of p(x)). b) If the cost function of producing x action gures is C(x) = 0. 004x2 10.125 x + 5000 Determine the profit function P(x). c) How many action gures should the company set as a sales target each month in order to maximize profit? d) At what sale price could the company expect to sell the action gures for maximum prot (from c)