Avoiding the Overfilled Can

Warm-up:

Upper and Lower bounds
(care of thirdspacelearning.com)

The Problem:

A production line fills 9.0 litres of paint into a cylindrical can of diameter 30cm.

What is the maximum (and minimum) height of the paint in the filled can?

-/|\-

Solution

The paint can is cylindrical and the formula for the volume, $V$, of cylinder is $$V=\pi r^{2}h$$ where the radius, $r=15\mathrm{cm}$ and $V=9.0\mathrm{lt}=9000{\mathrm{cm}}^3$.

Which leads to

But this question is really about bounding (upper and lower) because it asks for the maximum height of the paint. To get the maximum height we need to recalculate with the upper bound of the volume, $V_{\text{upper}}=9.05\mathrm{lt}=9050{\mathrm{cm}}^3$ and the lower bound of the radius, $r_{\text{lower}}=\frac{29.5}{2}=14.75\mathrm{cm}$.

Using these values we get

Similarly for

So $h=12.7\pm 0.5\mathrm{cm}$.

Further Reading:

Volume of 3-D Shapes
(care of mmerevise.co.uk)