A tank initially contains liters of water at time minutes

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

A tank initially contains 5050 liters of water at time t=0t=0 minutes. For 0≤t≤120\le t\le 12, water flows into or out of the tank at a rate (liters per minute) given by r(t)=5cos⁡ ⁣(πt6).r(t)=5\cos\!\left(\frac{\pi t}{6}\right). Positive values of r(t)r(t) correspond to water entering the tank, and negative values correspond to water leaving the tank. The volume of water in the tank at time tt is V(t)=50+∫0tr(u) du.V(t)=50+\int_{0}^{t} r(u)\,du. (All units are liters for volume and minutes for time.) (a) Determine the open intervals on 0<t<120<t<12 during which the volume V(t)V(t) is increasing and the intervals during which it is decreasing. Provide justification. (b) Find all times in 0<t<120<t<12 at which V(t)V(t) attains a local maximum or a local minimum. For each, state whether it is a maximum or minimum and give a clear reason for your classification. (c) Compute the exact maximum volume of water that the tank contains during the interval 0≤t≤120\le t\le 12 minutes.

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