Question

Let t be time in seconds and let r(t) be the rate, in gallons per second, that water enters a reservoir:
r(t)=700−40t.

a) For 0≤t≤30, when does the reservoir have the most water?
b) For 0≤t≤30, when does the reservoir have the least water?

Answer 1

if a(t) is the amount of water at time t, and the tank is initially empty,

da/dt = r(t) = 700-40t
a(t) = 700t-20t^2

a(t) is a parabola with vertex at t=700/40=17.5

So, that's the time of max water.

For minimum, find whether a(30) is positive or negative. Recall that a(0) = 0.