# One Surprising Way Calculus Can Help You Have A Better Vacation – Predict How Long The Line Will Be.

Calculus Level 4

You find yourself running a ski resort in Lake Tahoe, which is not the worst thing to have happened to your life. Frequently, you find long lines at the chairlift building up around midday which cause people to stay away from the mountain. In order to boost attendance, you're going to try and make the time spent waiting for the ski lift more entertaining. To be able to plan for this you need an estimate of how many people will be on the line.

For simplicity, your resort has just one run and one chairlift (maybe that is the real reason people stay home?). The ski run has a length of $$l_{run} = 4000$$ m, and all skiers have an average speed of $$v_{ski} = 8$$ m/s during their descent. The ski lift has a speed $$v_{lift}=1$$ m/s, has chairs spaced every $$l_{spacing} = 40$$ m, and each chair can hold $$g = 6$$ people. Finally, there is a total of $$N_T = 1000$$ people on the mountain.

Question: At steady state (when the average length of the line becomes constant), how many people do you expect to be waiting on the line?

Details and assumptions

• The length of the chairlift is the same as the length of the run, i.e. $$l_{lift} = l_{run}$$.
• As soon as people get off the lift at the top they begin skiing straight down the mountain.
• As soon as people get to the bottom of the run they get on the line for the chairlift.
• The effective length of the ski run is always $$l_{run}$$, no matter how long the line becomes.
• People wait for the chairlift in one single file line.
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