Phew! Not too steep! Once inside the water tower, Zachary easily defeats the kidnapper in hand-to-hand...

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Phew! Not too steep! Once inside the water tower, Zachary easily defeats the kidnapper in hand-to-hand combat, but getting the princess outside the tower will be tricky, since there is no ladder returning to the trap door. He will have to unlock the main doors by figuring out the complicated password before the tower fills with water and they both drown! Zachary finds a riddle by the doors, giving him the code he needs... if he can solve it in time! The riddle reads: 12) Use any method you know to determine the roots of these quadratic equations. (Round to 2 decimal places if necessary) Order the roots from least to greatest at the end to unlock the door! 12) Use any method you know to determine the roots of these quadratic equations. (Round to 2 decimal places if necessary) Order the roots from least to greatest at the end to unlock the door! a) x=4x+5 [6 marks] b) 2x+3x-4=0 c) x-49=0 Roots in order If you have done your job well, Zachary the Warrior and Princess Eva are safely returning home, perhaps a bit ragged and bruised, but alive to tell the story to their grandchildren. .oh yeah, they got married and lived happily ever after. Once inside the city, Zachary must rescue the princess who is being held prisoner in a water tower shaped like a cylinder with a cone-shaped roof. There is a mean guard at the bottom of the tower. Zachary decides to save his strength, and tries to bribe the guard with cash instead! 10) Zachary reaches into his pocket and pulls out $4.80 in nickels and quarters. He notices that he has 6 more nickels than quarters. How many of each coin does Zachary have? Solve for the answer using an algebraic method, and answer in a sentence. [4 marks] 8) Interpret the key features of the graph: a) What is the highest point that Zachary reaches through the air? b) How long does it take Zachary to reach his maximum height? c) For how long is Zachary in the air? [3 marks] 9) The factored form equation for Zachary's flight is given by y = ax(x-6). [3 marks] a) Substitute your vertex point into this equation to determine the "a" value, and round it to one decimal place. b) Find the standard form equation (y = ax2 + bx + c) for Zachary's flight. Vertex: ( height (m) 10 When Zachary the warrior finally reaches the Kingdom of Far Far away, there is a tall brick wall surrounding the city. Conveniently, there is a catapult nearby which Zachary uses to launch himself over the wall and into the city. His path through the air forms the parabola shown on the graph. 25 20- Height vs. Time 15- Time (s) 5 6 7) Identify the key features of the graph: [4 marks] x-intercepts and y-intercept: axis of symmetry (as an equation): . 6) The Kingdom of Far Far Away is 50 km from the river where Zachary started running. Does Zachary catch the kidnapper in time? (Circle your choice) YES NO [1 mark] If yes, how far away from the Kingdom are they when he catches up? If no, how far behind is Zachary when the kidnapper reaches Far Far Away? [1 mark] How long does it take Zachary to reach Far Far Away? [1 mark] b) Use substitution or elimination to find the hours and kilometres it will take for Zachary to catch the kidnapper. [4 marks] 5) Graph both Zachary and the kidnapper's distance equations on the grid below. Label the lines, and their point of intersection (give the coordinates). Hint: They should match your answer in #4. USE A RULER! [3 marks] 100 95 90 85 80 75 70 65 60 55 15 10 0 1 2 3 4 5 Distance vs. Time 8 9 10 11 12 13 14 15 16 17 18 19 20 time (hr) Safe or not, he made it up the tower, since he is, after all, a warrior. The ransom note read "If you wish to see Princess Eva alive, bring all the diamonds from the royal treasury to the Kingdom of Far Far Away". Zachary wastes no time. He heads for the Kingdom of Far Far Away, which involves crossing the Great River. He decides to take some measurements and use similar triangles to determine the width of the river. 3) Zachary can swim for a maximum of 75 metres before becoming too tired and drowning. Is he able to swim across the river? (hint: the ratio of two corresponding sides is the same for all pairs. Perhaps redrawing the triangles so they look similar will help.) [3 marks] 20m 50m 24m Once across the river, Zachary hopes to run and catch up to the kidnapper before he can reach the Kingdom of Far Far Away. 4) Zachary runs steadily at 10km/hr. His distance can be modelled by the equation: y = 10x, where y gives his distance (km) in relation to time (hours). The kidnapper runs at 5km/hr (since he's carrying a princess), and is already 30km away when Zachary begins his pursuit. a) Give the equation for the kidnapper's distance (y) in relation to time (x) In Slope-intercept form (y = mx + b): [2 marks] Standard Form (Ax + By + C = 0): Once upon a time a brave and mathematically-inclined warrior named Zachary set out on a quest to rescue the kidnapped Princess Eva. First, Zachary needed to get to the room of the stolen Princess to read the ransom note left behind. The Princess lived in a slanted tower with no stairs (for security purposes, ironically). Zachary plans to use his 17ft ladder to climb in a window 15ft up the tower's slanted wall. Zachary leans the ladder so that it just reaches the window. Now, ladders can be dangerous: too steep and they could topple backwards, too shallow and they could slide out from the wall! To be safe, the angle of elevation between the ladder and the ground should be between 73 and 78. 1) Will Zachary be safe on the ladder? Calculate the necessary angle(s) to justify your [3 marks] answer. 17 ft 15ft 86 2) Calculate how far from the base of the tower Zachary placed the ladder. (hint: You'll need the top angle first) [3 marks] 17 ft 15 ft 86 Phew! Not too steep! Once inside the water tower, Zachary easily defeats the kidnapper in hand-to-hand combat, but getting the princess outside the tower will be tricky, since there is no ladder returning to the trap door. He will have to unlock the main doors by figuring out the complicated password before the tower fills with water and they both drown! Zachary finds a riddle by the doors, giving him the code he needs... if he can solve it in time! The riddle reads: 12) Use any method you know to determine the roots of these quadratic equations. (Round to 2 decimal places if necessary) Order the roots from least to greatest at the end to unlock the door! 12) Use any method you know to determine the roots of these quadratic equations. (Round to 2 decimal places if necessary) Order the roots from least to greatest at the end to unlock the door! a) x=4x+5 [6 marks] b) 2x+3x-4=0 c) x-49=0 Roots in order If you have done your job well, Zachary the Warrior and Princess Eva are safely returning home, perhaps a bit ragged and bruised, but alive to tell the story to their grandchildren. .oh yeah, they got married and lived happily ever after. Once inside the city, Zachary must rescue the princess who is being held prisoner in a water tower shaped like a cylinder with a cone-shaped roof. There is a mean guard at the bottom of the tower. Zachary decides to save his strength, and tries to bribe the guard with cash instead! 10) Zachary reaches into his pocket and pulls out $4.80 in nickels and quarters. He notices that he has 6 more nickels than quarters. How many of each coin does Zachary have? Solve for the answer using an algebraic method, and answer in a sentence. [4 marks] 8) Interpret the key features of the graph: a) What is the highest point that Zachary reaches through the air? b) How long does it take Zachary to reach his maximum height? c) For how long is Zachary in the air? [3 marks] 9) The factored form equation for Zachary's flight is given by y = ax(x-6). [3 marks] a) Substitute your vertex point into this equation to determine the "a" value, and round it to one decimal place. b) Find the standard form equation (y = ax2 + bx + c) for Zachary's flight. Vertex: ( height (m) 10 When Zachary the warrior finally reaches the Kingdom of Far Far away, there is a tall brick wall surrounding the city. Conveniently, there is a catapult nearby which Zachary uses to launch himself over the wall and into the city. His path through the air forms the parabola shown on the graph. 25 20- Height vs. Time 15- Time (s) 5 6 7) Identify the key features of the graph: [4 marks] x-intercepts and y-intercept: axis of symmetry (as an equation): . 6) The Kingdom of Far Far Away is 50 km from the river where Zachary started running. Does Zachary catch the kidnapper in time? (Circle your choice) YES NO [1 mark] If yes, how far away from the Kingdom are they when he catches up? If no, how far behind is Zachary when the kidnapper reaches Far Far Away? [1 mark] How long does it take Zachary to reach Far Far Away? [1 mark] b) Use substitution or elimination to find the hours and kilometres it will take for Zachary to catch the kidnapper. [4 marks] 5) Graph both Zachary and the kidnapper's distance equations on the grid below. Label the lines, and their point of intersection (give the coordinates). Hint: They should match your answer in #4. USE A RULER! [3 marks] 100 95 90 85 80 75 70 65 60 55 15 10 0 1 2 3 4 5 Distance vs. Time 8 9 10 11 12 13 14 15 16 17 18 19 20 time (hr) Safe or not, he made it up the tower, since he is, after all, a warrior. The ransom note read "If you wish to see Princess Eva alive, bring all the diamonds from the royal treasury to the Kingdom of Far Far Away". Zachary wastes no time. He heads for the Kingdom of Far Far Away, which involves crossing the Great River. He decides to take some measurements and use similar triangles to determine the width of the river. 3) Zachary can swim for a maximum of 75 metres before becoming too tired and drowning. Is he able to swim across the river? (hint: the ratio of two corresponding sides is the same for all pairs. Perhaps redrawing the triangles so they look similar will help.) [3 marks] 20m 50m 24m Once across the river, Zachary hopes to run and catch up to the kidnapper before he can reach the Kingdom of Far Far Away. 4) Zachary runs steadily at 10km/hr. His distance can be modelled by the equation: y = 10x, where y gives his distance (km) in relation to time (hours). The kidnapper runs at 5km/hr (since he's carrying a princess), and is already 30km away when Zachary begins his pursuit. a) Give the equation for the kidnapper's distance (y) in relation to time (x) In Slope-intercept form (y = mx + b): [2 marks] Standard Form (Ax + By + C = 0): Once upon a time a brave and mathematically-inclined warrior named Zachary set out on a quest to rescue the kidnapped Princess Eva. First, Zachary needed to get to the room of the stolen Princess to read the ransom note left behind. The Princess lived in a slanted tower with no stairs (for security purposes, ironically). Zachary plans to use his 17ft ladder to climb in a window 15ft up the tower's slanted wall. Zachary leans the ladder so that it just reaches the window. Now, ladders can be dangerous: too steep and they could topple backwards, too shallow and they could slide out from the wall! To be safe, the angle of elevation between the ladder and the ground should be between 73 and 78. 1) Will Zachary be safe on the ladder? Calculate the necessary angle(s) to justify your [3 marks] answer. 17 ft 15ft 86 2) Calculate how far from the base of the tower Zachary placed the ladder. (hint: You'll need the top angle first) [3 marks] 17 ft 15 ft 86