MPM2D1 Culminating Activity Root of Fun Theme Park

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Margaret Harrell
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1 MPMD Culminating Activity Root of Fun Theme Park The MPMD class was 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: The Origin of Fun: the entrance area to the park Quadratics in Motion: rides and games based on quadratic relations Towers of Trigonometry: rides and games based on trigonometry and triangles 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. Because of your vast mathematical knowledge, you have been assigned the creating solutions to verify the calculations and information each group provides. As a result you must carry out each of the following assignments in addition to the people who have been assigned to them.. As the students were approaching the park, they noticed a huge tower that just being completed. Lucas and Jacob were part of the group responsible for looking at advertising. They couldn t help but to think, 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. These materials included: pencil, paper, eraser, calculator, measuring tape, a clinometer (a tool used to measure vertical angles). They walked through the park until the reached the shadow of the tower. They looked up and couldn t believe how high it was. a. Explain how the height of the tower might be calculated with the materials they have using similar triangles. Be sure to include diagrams, formulas and examples in your explanation. Students have complete freedom to pick a method of similar triangles to determine the height. The expectation is that many of the students would use shadows to create similar triangles (at least that is the hope with the emphasis being placed on I being a sunny day). However, some may use the pencil and stretch it out with their arm to create similar triangles. The explanations should include a detailed diagram, clear step by step instructions making use of terminology appropriately and even examples showing the calculations to be used to determine the height.

2 b. 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 use the materials they have and trigonometry to determine the height of the tower. Students have complete freedom to pick a method of using trigonometry to determine the height. The expectation is that many of the students would use the clinometer to measure the angle of elevation and the length of the shadow on the ground. Some students may measure an angle of elevation move a measured distance closer and then measure the angle of elevation again. They could then use the Sine Law to determine the measure of a side in the triangle followed by using the sine to calculate the height in the right angled triangle. The explanations should include a detailed diagram, clear step by step instructions making use of terminology appropriately and even examples showing the calculations to be used to determine the height. The question might be evaluated with the rubric displayed below. Category / Criteria Level (50%-59%) Level (60%-69%) Level 3 (70%-79%) Level 4 (80%-00%) Thinking Gathers information and uses decisionmaking skills to develop a process to solve the problem with limited success. Gathers information and uses decisionmaking skills to develop a process to solve the problem with some success. Gathers information and uses decisionmaking skills to develop a process to solve the problem successfully. Gathers information and uses decisionmaking skills to develop a process to solve the problem successfully and efficiently. Communicati on Explains how to determine the height of the tower using explanations, diagrams, examples, formulas and mathematical terminology with limited clarity. Explains how to determine the height of the tower using explanations, diagrams, examples, formulas and mathematical terminology with moderate clarity. Explains how to determine the height of the tower using explanations, diagrams, examples, formulas and mathematical terminology with considerable clarity. Explains how to determine the height of the tower using explanations, diagrams, examples, formulas and mathematical terminology with extensive clarity. Note: A student whose achievement is below Level (50%) has not met the expectations for this assignment or activity.

3 . A wooden ramp is being built to provide wheel chair access to the park. Stacey, Bob and Katherine drew out an initial plan where the ramp would span a horizontal distance of 9 m and a vertical distance of.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. Use similar triangles to determine the length of the vertical support? ABC ~ ADE (AAA ~) DE AD = BC AB x 5.5 =.7 9 x = 4.85 x =.65 Therefore the vertical support would be.65 m tall. b. Use trigonometry to determine the length of the vertical support? sina = O A sina =.7 9 A = sin.7 9 A = sina = O A x sin = sin = x.65 = x Therefore the vertical support would be.65 m tall.

4 3. The park initially planned to charge $8 for admission and expected to have 400 visitors a day. Allison and Hannah were assigned the task of analyzing the parks admission revenues. a. How much revenue would the park have for one day at the current price. ( ) ( ) Re # = venue of visitors price of admission = = 900 b. 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 $? ( ) ( ) Re venue = # of visitors price of admission = ( 400 ( 80) ) ( 8 + ( 050. )) = ( 40)( 9) = 060 c. 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. i. Write this equation in standard form, R = ax + bx + c. Show all of the steps leading to the final answer. R = ( x)( x) = x 640x 40x = 40x + 560x ii. What price should the park charge to maximize their revenue? R = 40x + 560x R = 40( x 4x) ( ) R = 40 x 4x R = 40( ( x 7) 49) R = 40( x 7) R = 40( x 7) + 60 Therefore, a maximum revenue of $ 60 occurs when x=7. Price = (7) Price = Price =.50 Therefore, the price that would maximize the park s revenue is $.50.

5 iii. What price range would produce a revenue over $0 000? R = 40x + 560x = 40x + 560x = 40x + 560x = 40x + 560x 800 ( x x ) ( x x ) 0 = = = x 4x + 0 Use the quadratic formula to solve for the roots. a =, b = -4, c = 0 b b 4ac x = ± a 4 ± (-4) 4()(0) x = () 4 ± x = x = 4 ± x = or x = x.4 or x.6

6 4. Brandon, Ericka and Tyler were assigned task of deciding upon a location of the first aid station. They started by placing a grid over a map of the park. They decided to place the entrance to the park at the origin. As a result, the locations of the main areas of the park were: Origin of Fun: (0, 0) Quadratics in Motion: (-7, 6) Towers of Trigonometry: (-, 4) The Food Equation: (5, 8) i. They decided the best place for the first aid station would be somewhere in the middle of the park. They decided to place the first aid station where the path joining The Origin of Fun and Towers of Trigonometry intersected with the path joining Quadratics in Motion and The Food Equation. What would be the coordinates of the first aid station? Provide mathematical support for your answer. Students could solve this graphically by plotting the points and then constructing the line segments that represent the two paths. First Aid Station: (-.00, 7.00) Tower of Trigonometry The Food Equation Quadratics in Motion First Aid Station 6 4 Origin of Fun Therefore the First Aid Station would be placed at (-, 7). The students could also create equations to model the two paths and then solve the resulting system of equations to determine the location.

7 ii. Does the location of the first aid station actually fall in the middle of the park? Provide mathematical calculations to support your answer. Calculate the midpoint of the path joining Origin of Fun to Tower of Trigonometry. midpoint = midpoint = midpoint = midpoint = x + x y + y, ( 7) + ( 5) 6 + 8,, (, 7) 4 Calculate the midpoint of the path joining Quadratics in Motion to The Food Equation. x + x y + y midpoint =, ( 7) midpoint =, midpoint = midpoint =, (, 7) 4 The First Aid Station could be considered as being in the middle of the park since it is in the middle of each diagonal path.

8 iii. In their research, Brandon found the first aid station must be located within 750 m of all major locations. If the scale on the map is square to 60 m, does the location of the first aid station meet this requirement? Provide mathematical calculations to support your answer. Map Distance d = ( x x ) + ( y y ) ( 0 ( ) ) ( 0 7) d = + d = ( ) + ( 7) d = + 49 d = 50 d = Actual Distance d = d = Therefore, the entrance, Origin of Fun, is about 44.3 m from the First Aid Station so it does meet this requirement.

9 5. A walking tour of the park is going to be offered as an option to visitors. Shelley, Hattie and Giffin are assigned the task of determining information about the tour. They place a grid over a map of the park and determine the locations of the main areas as follows: Origin of Fun: (0, 0) Quadratics in Motion: (-7, 6) Towers of Trigonometry: (-, 4) The Food Equation: (5, 8) A scale of square to 60 m is used for the grid. i. Determine the total distance that would be traveled on the tour that goes from The Origin of Fun, to Quadratics in Motion, to Tower of Trigonometry, to The Food Equation and then back to The Origin of Fun. Provide mathematical calculations to support your answer. Origin of Fun to Quadratics in Motion ( ) ( ) d = x x + y y ( 0 ( 7) ) ( 0 6) d = + d = ( 7) + ( 6) d = d = 85 d = Quadratics in Motion to Tower of Trigonometry ( ) ( ) d = x x + y y (( 7) ( ) ) ( 6 4) d = + d = ( 5) + ( 8) d = d = 89 d = Tower of Trigonometry to The Food Equation ( ) ( ) d = x x + y y (( ) 5) ( 4 8) d = + d = ( 7) + ( 6) d = d = 85 d =

10 The Food Equation to Origin of Fun ( ) ( ) d = x x + y y d = ( 5 0) + ( 8 0) d = ( 5) + ( 8) d = d = d = Calculate the total distance. d d = = Actual Distance d d = = Therefore, the walking tour would travel about 38.4 m.

11 ii. What shape is formed by the path traced in the walking tour? Provide mathematical calculations to support your answer. From the previous question we know the opposite sides are equal in length so the shape is either a rectangle or a parallelogram. Calculate the slope of each path. Origin of Fun to Quadratics in Motion slope m y y = = x x Quadratics in Motion to Tower of Trigonometry slope m y y = = x x 4 6 ( 7) 8 5 Tower of Trigonometry to The Food Equation slope m y y = = x x The Food Equation to Origin of Fun slope m y y = = x x The opposite sides are also parallel. The adjacent slopes are not negative reciprocals of each other so the shape is a parallelogram.

12 6. Doug, Marilyn and Tran are to provide advice on the suppliers for hamburgers for the food court. Isosceles Foods charges a delivery fee of $5.50 and $.0 for each hamburger. Scalene Wholesale charges a delivery fee of $.30 and $.7 for each hamburger. a. Determine the cost of purchasing 00 hamburgers from each company. Isosceles Foods Cost = ( 0. ) Cost = Cost = Scalene Wholesale Cost = ( 7. ) Cost = Cost = b. Determine the cost of purchasing 000 hamburgers from each company. Isosceles Foods Cost = ( 0. ) Cost = Cost = Scalene Wholesale Cost = ( 7. ) Cost = Cost = c. If cost is the main concern, describe the circumstances under which each company should be selected. Let C represent the total cost of the hamburgers. Let h represent the cost of each hamburger. Î C = h Ï C =.30+.7h Substitute Î into Ï C = h h = h = 7. h. h = 0. 07h = = h h Therefore, the Scalene Wholesale company should be selected if 574 or fewer hamburgers are going to be ordered and the Isosceles Foods company should be selected if more that 574 hamburgers are being ordered.

13 7. A section of the roller coaster track is displayed in the diagram below. 5 h d Wally, Janet and Sunhil are responsible for the roller coaster design. a. Write the equation of the graph that would model the roller coaster track displayed in the diagram. vertex (7, ) step pattern:, 6, 0,... opens down Therefore, the equation to model this section of the roller coaster track is h = ( d 7) +.

14 b. A straight section of track will be used to join the point (5,6) to the existing track displayed in the diagram. Determine the slope of this section of straight track. slope m y y = = x x Therefore, the slope of this section of track is 3. c. The first part of the track is not displayed in the diagram. They plan on y = x 6x + using the equation. i. Change the equation into vertex form, y = a( x p) + q. y = x 6x + y = x 6x y = ( x 3) 9 + y = ( x 3) + 5 ii. h Sketch the missing part of the roller coaster and label the vertex and two other points on this section of the track d

MPM 2DI EXAM REVIEW. Monday, June 25, :30 am 10:00 am ROOM 116 * A PENCIL, SCIENTIFIC CALCULATOR AND RULER ARE REQUIRED *

MPM 2DI EXAM REVIEW. Monday, June 25, :30 am 10:00 am ROOM 116 * A PENCIL, SCIENTIFIC CALCULATOR AND RULER ARE REQUIRED * NAME: MPM DI EXAM REVIEW Monday, June 5, 018 8:30 am 10:00 am ROOM 116 * A PENCIL, SCIENTIFIC CALCULATOR AND RULER ARE REQUIRED * Please Note: Your final mark in this course will be calculated as the better