1. On a sunny and calm day in May, Damond goes on a hot air balloon...

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1. On a sunny and calm day in May, Damond goes on a hot air balloon ride. The hot air balloon launches from the top of a hill, which is located 563 meters above sea level and first it moves straight up and then straight down vertically (it does not move sideways). During the hot air balloon ride, Damond measures the velocity of the balloon and records it in the below table, where time, t, is measured in seconds and upward velocity, v(t), is measured in meters per second: t 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 v(t) 0 0.8 1.2 1.9 2.0 0.5 0 -0.3 -1.1 -0.7 -1.5 -1.3 -1.2 -0.2 0 (a) (2 marks) Find the Riemann sum of v(t) on [0, 420] with n = 14 using right endpoints. What does the Riemann sum represent, in terms of the hot air balloon? (Do not answer in terms of areas of the graph, but explain what the Riemann sum means in terms of the movement of the balloon.) (b) (2 marks) Use a Riemann sum with n = 4 and right endpoints, to estimate the average velocity (the average value of v(t)) during the first 2 minutes. Do you think your answer is an overestimate or an underestimate of the actual average velocity? If you make any assumptions about the data in the table, or about the balloon ride, please state those assumptions and why you think they are reasonable. (c) (2 marks) Use a suitable Riemann sum to estimate the maximum altitude above sea level that the balloon reaches during the flight. (d) (2 marks) Damond's friend Noor (who also was on the ride) says she remembers that the hot air balloon came back to the ground after 7 minutes. Considering the measurements in Damond's table as well as your answer to part (a), does that seem reasonable? Why/why not? (e) (2 marks) Damond's friend Kali (who watched from the ground) claims that at times t = 60 and t = 360, the balloon was at the same height above ground. Considering the measurements in Damond's table, does that seem reasonable? Why/why not? 1. On a sunny and calm day in May, Damond goes on a hot air balloon ride. The hot air balloon launches from the top of a hill, which is located 563 meters above sea level and first it moves straight up and then straight down vertically (it does not move sideways). During the hot air balloon ride, Damond measures the velocity of the balloon and records it in the below table, where time, t, is measured in seconds and upward velocity, v(t), is measured in meters per second: t 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 v(t) 0 0.8 1.2 1.9 2.0 0.5 0 -0.3 -1.1 -0.7 -1.5 -1.3 -1.2 -0.2 0 (a) (2 marks) Find the Riemann sum of v(t) on [0, 420] with n = 14 using right endpoints. What does the Riemann sum represent, in terms of the hot air balloon? (Do not answer in terms of areas of the graph, but explain what the Riemann sum means in terms of the movement of the balloon.) (b) (2 marks) Use a Riemann sum with n = 4 and right endpoints, to estimate the average velocity (the average value of v(t)) during the first 2 minutes. Do you think your answer is an overestimate or an underestimate of the actual average velocity? If you make any assumptions about the data in the table, or about the balloon ride, please state those assumptions and why you think they are reasonable. (c) (2 marks) Use a suitable Riemann sum to estimate the maximum altitude above sea level that the balloon reaches during the flight. (d) (2 marks) Damond's friend Noor (who also was on the ride) says she remembers that the hot air balloon came back to the ground after 7 minutes. Considering the measurements in Damond's table as well as your answer to part (a), does that seem reasonable? Why/why not? (e) (2 marks) Damond's friend Kali (who watched from the ground) claims that at times t = 60 and t = 360, the balloon was at the same height above ground. Considering the measurements in Damond's table, does that seem reasonable? Why/why not?