A mountain ski resort tracks both snowfall depth and snow-grooming removal over a…
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A mountain ski resort tracks both snowfall depth and snow-grooming removal over a 24-hour period.
• The depth of newly fallen snow on the main slope, in centimeters, is S(t) at hours 0≤t≤24 after midnight.
• The amount of snow removed by grooming machines, in centimeters, is G(t) over the same interval.
Both functions are differentiable. The resort provides the following information:
1. For 0≤t<6, S′(t)=2.6 and G′(t)=0 (steady snowfall, no grooming).
2. For 6≤t<12, S′(t)=1.8 and G′(t)=0.9 (lighter snowfall, grooming begins).
3. For 12≤t<18, S′(t)=0.5 and G′(t)=1.4 (minimal snowfall, heavy grooming).
4. For 18≤t≤24, S′(t)=0 and G′(t)=1.1 (no snowfall, continued grooming).
Time is measured in hours after midnight, and all rates are in centimeters per hour.
(a) For each of the four time intervals listed, state the net rate of change of snow depth on the slope, defined as S′(t)−G′(t), and include units.
(b) Identify every interval during which the depth of snow on the slope was decreasing. Justify your answer based on the sign of the net rate.
(c) Calculate the total amount of snow added to the slope from snowfall alone during the 24-hour period.
(d) The resort claims that between midnight and 24:00, grooming removed snow at an average rate that exceeded the average snowfall rate. Using appropriate average rate calculations, confirm or refute this claim with clear justification.
(e) Suppose resort safety regulations require that the instantaneous removal rate must never exceed twice the instantaneous snowfall rate. Determine whether this safety condition was violated at any time during the 24-hour period, and explain your conclusion.