As a die-hard Steelers fan, Juan Jr. Jr. is interested in everything Ben Roethlisberger. He is watching a Steelers Game on Sunday night with his family and he notices how amazing Ben’s throwing is. Then all of a sudden, Great Grandpa Juan collapses and falls to the ground. As he is taking his last breathe, mathematician, Great Grandpa Juan asks Jr. to find the IRC of Roethlisberger’s winning pass at 7 seconds (t=7). A few weeks later, Juan Jr. Jr. decides to fulfill his great grandfather's last wish and finds the IRC at t=7. These are the steps he takes
Time (t) in seconds
|
height (ft)
|
0
|
6
|
1
|
21
|
2
|
44
|
3
|
53
|
4
|
45
|
5
|
33
|
6
|
26
|
7
|
19.5
|
8
|
12
|
9
|
5
|
Average Rate of Change
ARC: (y2-y1)/(x2-x1)
(7,19.5) and (9,5)
(5-19.5)/(9-7)= -7.25 (R)
(7, 19.5) and (8,12)
(12-19.5)/(8-7)= -7.5 (R)
(7, 26) and (6,26)
(27.5-19.5)/(6-7)= -8 (L)
(7, 19.5) and (5,33)
(33-19.5)/(5-7)= -6.75 (L)
In order to find the ARC I had to use the slope formula. Juan Jr. Jr. used the point at for which he was trying to find the IRC, (7,19.5), and then used other points from the chart. From the calculations he saw that the ARC was negative on both the right and left side of t=7. From the right side the numbers are getting smaller and smaller and from the left side the numbers are getting bigger and bigger.
Instantaneous Rate of Change
To find the IRC he drew a tangent line to t=7 on the graph. He picked a point on this tangent line in order to calculate the IRC. He chose the point (7.5,17.3). Using the original point (7,19.5) and this new point Juan Jr. Jr. was able to find the IRC.
(7,19.5) and (7.5, 17.3)
(19.5-17.3)/(7-7.5) = -4.4
From this calculation the IRC is concluded to be 4.4sec/ft. The IRC is the same thing as saying the derivative of the slope at a point. The ARC were getting smaller and smaller or larger and larger from both sides to each other. So this is the slope or Instantaneous rate of change at 7 seconds, meaning that is how fast the football is going at that certain point.
Hey Joti! I really enjoyed reading your scenario. I liked how you brought in Juan Jr. Jr. from our group activity! He has such an interesting last wish before he dies. Your calculations look correct from what I can see. Good job on this very creative blog post!
ReplyDeleteHey Joti! This was a very interesting blog post. You choose an interesting application and wrote the blog in a creative way. Your work was also very clear and from what i can tell, correct. Good job
ReplyDeletejoti,
ReplyDeleteummm...first of all, i LOVE that you are continuing the tale of juan jr. jr. from your group activity story! that's awesome!
in addition to being creative with your storytelling, you also did a wonderful job of accurately calculating the IRC. wonderfully organized and explained! the only thing that i may have added in your concluded comments would have been an explanation of what 4.4 means (i.e. that the ball was decreasing at a velocity rate of 4.4 feet per second) excellent job!
professor little