Having a little trouble with an optimization question:

A half-spherical raindrop of diameter 1cm is sitting on a picnic table. It is evaporating
at a rate of 1cm^3 per 10 min. How fast is the rop's circular footprint shrinking when
the diameter is half its original width?

So what I think needs to be found is $\dfrac{\mathrm{d}D}{\mathrm{d}t}$ and I know that the volume of a half sphere is $\dfrac{4 \pi r^3}{6}$.

So what I have to do is differentiate the equation and then half it...? How do I do this?

  • $\begingroup$ This isn't an optimization problem, that would mean that you are trying to find the maximum/minimum value of something. This problem is called a "related rates problem". $\endgroup$
    – Spencer
    Jun 22, 2014 at 3:10

1 Answer 1


You need to relate the rate you know, $dV/dt$, to the rate you want, $dA/dt$. We can do this using the chain rule,

$$ \frac{dV}{dt} = \frac{dV}{dr} \frac{dr}{dt} = \frac{dV}{dr} \frac{dr}{dA} \frac{dA}{dt}, $$

everything in this equation is known except $dA/dt$ which you can solve for after plugging the other information in.

  • $\begingroup$ See how far you can get with this hint. If you need more assistance update your question showing me where you are stuck. $\endgroup$
    – Spencer
    Jun 22, 2014 at 3:09
  • $\begingroup$ Thanks, I got it, also It was to find dA/dt because its asking about the footprint of the sphere on the table getting smaller. I ended up getting -0.1cm^2/min $\endgroup$
    – Panthy
    Jun 22, 2014 at 3:20
  • $\begingroup$ Cool, glad I could help! I did wonder if they wanted an area to describe the footprint. I'll edit my answer to change $D\rightarrow A$. $\endgroup$
    – Spencer
    Jun 22, 2014 at 3:23

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