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

What you don't want is the value of r(t) becoming negative. Surely that would represent water escaping the reservoir.

How big can (t) get before water actually starts escaping the reservoir?

Essentially, to figure this out r(t) would have to be equal to 0. 

700 - 40t = 0

40t=700

t=700/40=17.5

So the first answer is 17.5 seconds. After this amount of time has elapsed the reservoir will start to lose water as r(t) would become negative.



The reservoir had the least amount of water in it before it was being filled. That was when t=0. The volume of water in the reservoir wasn't negatively impacted as not enough water had escaped it during the 17.5 to 30 second period.