It is worth noting, in my opinion, that my optimisation example illustrates a general exam strategy that I believe is under-utilised, which is:
Keep your eye on your goal.
In this question, my goal was the ratio
![](https://latex.codecogs.com/png.latex?\bg_white r:h)
, and while that can be found by finding
![](https://latex.codecogs.com/png.latex?\bg_white r)
and
![](https://latex.codecogs.com/png.latex?\bg_white h)
and then dividing them, I am not actually required to find
![](https://latex.codecogs.com/png.latex?\bg_white r)
or
![](https://latex.codecogs.com/png.latex?\bg_white h)
.
In forming equation (2), I recognised that I was planning to use it to eliminate
![](https://latex.codecogs.com/png.latex?\bg_white r)
or
![](https://latex.codecogs.com/png.latex?\bg_white h)
from equation (1). Thus, I expressed (2) in the form that suited simplifying (1), rather than making
![](https://latex.codecogs.com/png.latex?\bg_white h)
the subject and adding an extra line of algebra into the substitution step to get to the equation linking
![](https://latex.codecogs.com/png.latex?\bg_white A)
to
![](https://latex.codecogs.com/png.latex?\bg_white r)
.
As I showed later, rearranging equation (2) gave me a form for
![](https://latex.codecogs.com/png.latex?\bg_white r:h)
where the only variable present was
![](https://latex.codecogs.com/png.latex?\bg_white r^3)
.
Thus, I worked with the stationary point being at
![](https://latex.codecogs.com/png.latex?\bg_white r^3=\frac{V}{2\pi})
. I knew that I could take the cube root if I needed
![](https://latex.codecogs.com/png.latex?\bg_white r)
but recognised also that knowing
![](https://latex.codecogs.com/png.latex?\bg_white r^3)
was likely to be sufficient. This also told me that using equation (2) to back-substitute and find an exact form for
![](https://latex.codecogs.com/png.latex?\bg_white h)
was likely to be unnecessary.
In other words, I didn't stop to find an explicit form for
![](https://latex.codecogs.com/png.latex?\bg_white h)
or
![](https://latex.codecogs.com/png.latex?\bg_white r)
as neither of them was my goal.
In doing so, I avoided:
This is a strategy that can save you time and possibly also avoid losing marks from mistakes in working that isn't actually needed anyway.