224c224
< \vspace*{-3.0cm}
---
> \vspace*{-3.5cm}
260c260
< (There is no picture in the solutions.)
---
> % (There is no picture in the solutions.)
262a263,266
> Do {\textbf{not}} use a separate letter for any velocity, rate of
> change, etc.
> These are derivatives of quantities you should have
> already named with letters.
268c272,274
< Relate the variables.
---
> Relate the variables, giving a relation that is valid for all
> values of time, not just the time for which the information in the
> problem is given.
272,273c278,281
< Differentiate the relation, getting an equation involving
< quantities and their derivatives.
---
> Differentiate the relation in step~(\ref{5512_Relate})
> {\textbf{with respect to time}},
> getting an equation involving
> quantities from step~(\ref{5512_Picture}) and their derivatives.
279c287,290
< Put known values in the equation above,
---
> Now, and only now, restrict to the particular time given in the problem.
> (In many solutions, this will be ``put $t = t_0$'', ``put $t = 0$'',
> or similar.)
> Put the known values in the equation above,
288,289c299,300
< You will need the volume of a sphere of radius~$r$.
< It is $\dfrac{4}{3} \pi r^3$.
---
> You will need the volume $V$ of a sphere of radius~$r$.
> It is $V = \dfrac{4}{3} \pi r^3$.
303a315
> \noindent
331a344,350
> 
> \smallskip
> 
> This must be done for arbitrary time, not just for the particular
> time at which the information is given.
> Otherwise, in the next step, you are just differentiating constants,
> and you will get zero.
347a367
>