This activity makes up 20% of your final grade. It is marked out of 86.

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1 MFM2P/MPM2D Name: Culminating Activity Root of Fun Theme Park This activity makes up 20% of your final grade. It is marked out of 86. The grade 10 math class took a trip to the Root of Fun Theme Park that was being built in the area. The park is designed to have four main areas: 1. The Origin of Fun : the entrance area to the park 2. Quadratics in Motion : rides and games based on quadratic relations 3. Towers of Trigonometry : rides and games based on trigonometry and triangles 4. Food Equation : food court area As part of their trip, they must use their mathematical knowledge to provide assistance with the design of the park. You have been assigned the task of creating solutions to verify the calculations and information each group provides. As a result, you must carry out each of the following tasks with the people who have been assigned to them. Task 1 As the students were approaching the park, they noticed a huge tower that was just being completed. Lucas and Jacob were part of the group responsible for looking at advertising. They couldn't help but to think that one of the main attractions of the park would be the ride involving this tower. It was a bright, sunny day. As they got off the bus, they collected the mathematical materials provided by their teacher: pencil, paper, eraser, calculator, measuring tape, a clinometer (a tool used to measure vertical angles). They walked through the park until they reached the shadow of the tower. They looked up and couldn't believe how high it was. 1. Explain how the height of the tower might be calculated from the materials they have, using similar triangles. You might want to sketch a picture of the tower as part of your answer. ( 4 marks )

2 2. If they are going to advertise the height of the tower in a brochure that is being created, they want to be sure of their answer. Describe how they could determine the height of the tower using measurements and trigonometric ratios. Include a diagram with your solution. ( 4 marks ) 3. A wooden ramp is being built to provide wheelchair access to the park. Stacey, Elias and Keiko drew out an initial plan where the ramp would span a horizontal distance of 9 m and a vertical distance of 2.7 m. However, they decided the ramp would need extra support. As a result, they decided to place an extra vertical support 5.5 m from the point where the ramp meets the ground. a. Sketch a diagram of the ramp. ( 3 marks )

3 b. Use similar triangles to determine the length of the vertical support. ( 4 marks ) c. Use trigonometric ratios to determine the length of the vertical support. ( 4 marks )

4 Task 2 The park had initially planned to charge $8 for admission and expected to have 2400 visitors a day. Allison and Juan were assigned the task of analyzing the park's admission revenues. 1. How much revenue would the park have for one day at the current price? ( 3 marks ) 2. They have been provided with some market research that shows for every $0.50 the admission price is raised, the park will have 80 fewer visitors. How much would the park revenue be if park raised their admission price by $1? ( 4 marks ) 3. After a few calculations, Allison and Hannah realize the park will make more money if they raise the price of admission. However, they also understand that there must be a limit to how much the park can charge. As a result, they model the situation with the equation, R = ( x)( x), where R represents the revenue from sales and x represents the number of price increases. a. What form is the revenue equation in? How do you know? ( 2 marks )

5 b. Convert this equation to standard form, R = ax 2 + bx + c. ( 4 marks ) c. What price should the park charge to make the most revenue? ( 5 marks ) d. What price would produce a revenue of zero dollars? How do you know? ( 2 marks )

6 Task 3 Brandon, Ericka and Giovanni were assigned the task of deciding upon the location of the first aid station. The locations of the main areas of the park were: A Origin of Fun: (0, 0) B Quadratics in Motion: ( 7, 6) C Towers of Trigonometry: ( 2, 14) D The Food Equation: (5, 8) They decided to place the first aid station at the coordinates ( 1, 7). 1. Graph the park locations below. Label everything. ( 5 marks )

7 2. Does the location of the first aid station actually fall in the middle of the park? Provide mathematical calculations to support your answer. ( 4 marks ) 3. If Ericka walked from A to C and Giovanni walked from B to D, a. Determine the equation of the line of Ericka s path and Giovanni s path (one equation for each). ( 4 marks ) Ericka s Equation Giovanni s Equation

8 b. Where would their paths cross each other? Use substitution or elimination method to figure this out. ( 5 marks )

9 Task 4 A walking tour of the park is going to be offered as an option to visitors. Shelley, Hattie and Griffin are assigned the task of determining information about the tour. They use the same coordinates as the previous question: A Origin of Fun: (0, 0) B Quadratics in Motion: ( 7, 6) C Towers of Trigonometry: ( 2, 14) D The Food Equation: (5, 8) Every square on the graph equals 60 m in the park. 1. Determine the total distance that would be traveled on the tour that goes from A to B to C to D, and then back to A. Provide mathematical calculations to support your answer. ( 8 marks )

10 2. What shape is formed by the path traced in the walking tour? How do you know? ( 3 marks ) Task 5 Doug, Marilyn and Tran must provide advice about the hamburgers for the food court. Isosceles Foods Scalene Wholesale delivery fee of $52.50 delivery fee of $12.30 $1.10 for each hamburger $1.17 for each hamburger 1. Determine the cost of purchasing 200 hamburgers from each company. ( 3 marks ) 2. Determine the cost of purchasing 2000 hamburgers from each company. ( 3 marks )

11 3. If cost is the main concern, describe the circumstances under which each company should be selected. ( 2 marks ) Task 6 A section of the roller coaster track is displayed in the graph below.

12 Wally, Janet and Sunhil are responsible for the roller coaster design. 1. Label three important parts of the graph. ( 3 marks ) 2. Write the equation of the graph that would model the roller coaster track displayed in the diagram. Use vertex form. ( 3 marks ) 3. The first part of the track is not displayed in the diagram. They plan on using the equation y = x x Convert the equation into factored form, y = a(x p)(x q ). ( 4 marks )