\begin{homework}\
\label{homework:3}

\medskip
\exercise
The graph of $y=\sqrt{x}$ is rotated around the $y$-axis to form a
container into which liquid is being poured at the rate of 0.1 cubic units
per minute.  

(a) How many minutes before the container is filled to a height of 1?

(b) At that point ($h=1$), how fast is the water level rising?

\smallskip
\exercise
Same as Problem 1, but with the graph of $y=x^4$ rotated around the
$y$-axis.

\smallskip
\exercise
Sketch the graph of $x=\tan(\pi y/20)$.  This graph is rotated around
the $y$-axis to form a vat (the units are feet).  Liquid is poured into the
vat in such a way that the height of liquid is increasing steadily at 0.2
ft/sec.  At what rate (in ft${}^3$/sec) is liquid being poured when the
height is 5 ft?

\smallskip
\exercise
A hemispherical vat of radius 2 meters is being filled with a liquid,
starting at time $t=0$ sec, in such a way that the depth of the liquid
increases at the steady rate of 4 cm/sec.  What is the rate of flow of the
liquid (in m${}^3$/sec) after $t$ seconds (expressed in terms of $t$)?

\smallskip
\exercise
Same as Problem 4, except that the vat is obtained by rotating around
the $y$-axis the bottom half of the ellipse $x^2\!/9+y^2\!/4=1$.

\smallskip
\exercise
With the vat in Problem 5, suppose now that it is being filled at the
steady rate of 0.15 m${}^3$/sec.  Find the rate at which the depth of the
liquid in the container is increasing at the moment when its depth is 0.5
m.

\smallskip
\exercise
Suppose that the tetrahedron in Problem 9 of the previous chapter is
situated upside down with its vertex at the origin and its base perpendicular
to the $y$-axis at $y=H$.  If the tetrahedron is being filled with water,
find the ratio of $dV/dt$ to $dh/dt$ at the instant when the water has
depth $h$ at the deepest point.  Express your answer in terms of $h$, $H$,
and $s$.

\end{homework}

