After a long day of super h-core skiing, Luke and Kelsey disagree on what run to take as their last run. Kelsey wants to ski Cascade, the purple run, and Luke wants to ski Funnel to Funnel Bypass, the blue run:

(Thanks to http://aspensnowmass.com/images/dyn/trailmaps/jpg-High/sn.html for the trail map.)

Since these runs are at right angles, the two math whizzes got a wonderful idea:

"Let's turn this into a fun calculus problem," said Kelsey.

"Yeah! Since these runs are at a right angle, this can be a related rates problem, how exciting!" agreed Luke, thrilled to be combining his two most favorite things in the world: skiing and calculus.

"We can get the length of the runs by asking the ski patrol and then time ourselves going down them to find out speed and then use our speeds to find the rate at which the distance between us is increasing at a certain time," Kelsey was about to pee her pants.

Luke's run (blue) is L, has a length of 1.325 miles, and Kelsey's run (purple) is K, has a length of 2.12. The distance between them is X (red). Luke takes off from the top and reaches the bottom 3 minutes after leaving, and Kelsey takes off at the same time and reaches the bottom 3.5 minutes after leaving. Find the rate at which the distance between them (X) is changing at exactly 1.5 minutes.

Luke and Kelsey take their runs and meet in the parking lot with the data to figure out their problem:

"First we need to find our speed by dividing the lengths of the runs by the time it took us to get down them."

Kelsey's speed:

Luke's speed:

"Then we need to use our known speeds to find our distances from the top at 1.5 minutes. "

Kelsey's distance (K): 

Luke's distance (L):

The general equation, to find the length of X is:

The differential equation, to find rate of change of the length of X is:

and since we are trying to find dX/dt, we need to isolate it, which make the equation:

and now all we need to do is substitute! 

The two of them met up at the bottom, and discussed their run:

"This means that at 1.5 minutes we were moving apart at a rate of  .663 miles per minute."

Kelsey said " How cool is it that we calculated the rate at which we were moving apart?"

"I always want to know my proximity to my ski partner, so it is great that we can do this whenever we want to!" responded Luke

And that ended the sick day on the mountain.

Luke Reflection: I did most of the filming and editing for this project. I learned a lot within the film and video aspects of Web 2.0, like iMovie and Youtube. I also learned a lot about Aspen, and talked to most of their PR department! This was helpful because it reminded me that if you try hard enough, you can likely find what your looking for. Big thanks to Aspen for providing us with the lengths of the runs we needed. I also learned a lot of new songs from being on hold! I think this project worked well because each of us had very similar goals and we all shared work quite equally. Each of us understood each role we needed to play and got our individual job done. We also had a lot of fun creating this Related Rates Problem, which is the most important part! 

Gus Reflection: I mostly helped out where it was needed, I did some calculations, though Kelsey did the majority of that, and Luke did the editing. I think that I learned a lot about working in a group. I am not usually a very organized person, but as we approached the deadline, I learned that it is important to keep it together. In a group organization and team work is so important, this was a useful lesson in discipline.

Kelsey Reflection: I did a lot of the calculations and writing of the problem with the help of Gus. I learned that it is definitely important to stay on top of things. I wrote the frame work for the problem early so that when we got the number all we had to do was plug them in. I mean substitute. Using equation editor ended up taking a while so I'm glad we had everything else done. Luke did all of the video editing stuff, mostly because he's the only one that knows how and Gus helped us both. I think the work load evened out between everybody. I was really excited to find that my environmental science class is related to calculus. We are learning about climate change and we just looked at a graph that shows the atmospheric CO2 at an observatory in Mauna Loa, Hawaii. The graph has an increasing trend and we were talking about predicting what the CO2 levels would be in the future. Then I started thinking, that we would have to know the rate at which the CO2 levels are changing because I don't think they are constant. I am making this assumption based off of statistics that I have about temperature, which is directly related to CO2. From 1906 to 2005 the temperature increased about .74˚C, but from 1956 to 2005 the increase was .65˚ C (The Physical Change behind Climate Change). This means that most of the increase over the past hundred years happened in the last 50 years. This means that the rate at which the temperature and CO2 levels were changing 50 years ago was less than the rate at which they are changing now. We know that the levels are increasing. We would need to know the rate at which they are increasing (derivative) and maybe even the rate at which the rate of increase is increasing (second derivative) to be able to accurately predict the future for CO2 levels.