A municipal water reservoir is being filled and drained simultaneously by several pipes
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A municipal water reservoir is being filled and drained simultaneously by several pipes. The volume of water in the reservoir, measured in millions of gallons, is modeled by a differentiable function V with derivative
V′(t)=2.4sin(0.4t)−0.6t,
where t is the number of days since monitoring began.
City planners wish to understand when the water level is rising or falling and how this information should guide scheduling of maintenance that requires low water levels.
(a) Determine all critical points of V for 0≤t≤15. Show analytical work that supports your findings.
(b) Create a sign chart for V′(t) on the interval 0≤t≤15 and use it to identify each subinterval on which the reservoir volume is increasing and each subinterval on which it is decreasing.
(c) Justify the classification (local maximum, local minimum, or neither) of each critical point identified in part (a) using the First Derivative Test.
(d) Maintenance crews need a 24-hour period during which the water level is steadily decreasing. Based on your analysis, propose an appropriate time window within the first 15 days and justify your recommendation.