Question

The lifts at Purgatory open at 9 am (t=0) and close at 4 pm (t=7). On a certain run, the minimum depth of snow for safe skiing is 40 cm. Purgatory wants to keep the run open from 9 am to 4 pm, but also maintain the illusion of natural snow. So, they will run the snow making machine at the top of each hour, but run it the minimum length of time necessary to maintain a depth of snow of at least 40 cm until 4 pm. For practical reasons, they will set the machine to run the same length of time each hour- that is they will not wait until it's needed to run it on, but rather proactively run the machine the same length of time each hour.

The snow machine:

-makes snow at a rate of 4 cm/hour

-can only be set to run a whole number of minutes: 2 minutes, 3 minutes, etc.

The snow depth:

-decreases at a rate proportional to the depth of snow (more now attracts more skiers). Purgatory has determined from experience that the constant of proportionality is is 0.03.

-It is 41 cm at 9 am.

Determine the minimum time setting for the snow machine to achieve the desired result. What will the depth of the snow be at 4 pm with this setting?

(a) The differential equation you use to model this scenario.

(b) The time value you determine.

(c) The depth of snow at 4 pm.

  

Answer 1

At first I've derived the differential equation and formula needed to solve this problem and then I explained all of them in words. Finally I wrote the required answers.