Figure N4–23SOLUTION: Liquid flowing in at a constant rate means the change in volume is
constant per unit of time. Obviously, the depth of the liquid increases as t does,
so h′(t) is positive throughout. To maintain the constant increase in volume per
unit of time, when the radius grows, h′(t) must decrease. Thus, the rate of
increase of h decreases as h increases from 0 to a (where the cross-sectional area
of the vessel is largest). Therefore, since h′(t) decreases, h′′(t) < 0 from 0 to a and
the curve is concave down.
As h increases from a to b, the radius of the vessel (here cylindrical) remains
constant, as do the cross-sectional areas. Therefore h′(t) is also constant,
implying that h(t) is linear from a to b.
Note that the inflection point at depth a does not exist, since h′′(t) < 0 for all
values less than a but is equal to 0 for all depths greater than or equal to a.