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Question

During a severe rainstorm water enters a catch basin at a rate of 𝑅 𝑡 = −𝑡^2+ 4𝑡 (gallons/hour) where x is time in hours over the course of four hours 0 ≤ 𝑡 ≤ 4. Water leaves the catch basin via a drain at the rate of𝐿(𝑡)= 2𝑡. Find the time, 𝑡, 0≤𝑡≤4, at which the water in the catch basin is decreasing at the greatest rate.

Answer 1

The net change in the amount of water in the basin is given by C(t) = R(t) - L(t) = -t^2 + 2t.

The maximum decrease in this function could be at either "end" of the spectrum (t=0 or t=4) or at a point in the middle where dC/dt = 0. Since dC/dt = -2t +2 =0 at t =1, we have to check t=0, 1, and 4.

If you do this, you'll find that at t=1 the rate is actually a maximum (*increasing* at the greatest rate). The minimum (*decreasing* at the greatest rate) is at t = 4.