Meteorologists track the depth of snow (in inches) on a mountain over a 24-hour…
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
Meteorologists track the depth of snow S(t) (in inches) on a mountain over a 24-hour period beginning at midnight. Between midnight and 6:00 a.m. snowfall is steady at 0.3 in/h. From 6:00 a.m. to noon, a storm increases the rate, and data suggest modeling the depth by S(t)=b(t−6)4+S(6) for 6<t≤12, where t is in hours and b is a positive constant. After noon no additional snow falls, and melting begins at a constant rate of 0.25 in/h for 12<t≤24.
(a) Using the description, express S(t) as a complete piecewise function on 0≤t≤24, introducing any needed parameters.
(b) Explain how the requirement that snow depth change smoothly rules out any jump discontinuity in S at both t=6 and t=12.
(c) Determine conditions on your parameters from part (a) that ensure S is continuous at t=6 and t=12.
(d) Assuming those conditions hold, justify that there must be a time between midnight and noon when the snow depth exceeded 5 inches, given that S(0)=2.0 in and S(12)=7.8 in.