Calculating Uncertainty: Where Am I Going Wrong?

[Taylor_1989]

Homework Statement

A histological slide contains darkly stained cells of two kinds-normal cells which are circular in shape and elliptical cells f the same area (which are pathological). An automated image processing sequence aims to identify and distinguish the individual objects and calculating their respective form factors.

The are of an ellipse is $A=\pi ab$( where a is the length of the semi-major axis, and b is the length of the semi-minor axis). The perimeter of an ellipse is quite difficult to calculate but a simply and fairly crude approximation is given by

$$P=2\pi \sqrt{\frac{a^2+b^2}{2}}$$

d) using the formulae above, calculate #f# for an ellipse and deduce its maximum and minimum values as $a$ ranges from $b$ to $3b$. Sketch $f$ as $a$ ranges from $b$ to $3b$

e)We assume that the perimeter of the ellipse given by the formula above is uncertain to about 5% of it true value bu the area is know exactly.
Derive the corresponding uncertainty in $f$, and use your sketch from part d) (or otherwise) to deduce the minimum value of the ratio of $a$ to $b$ which would enable a clear distinction to be made between normal and pathological cells?

Relevant Equations

Uncertainty equation
$$Y=X^n$$
$$\Delta Y = |n|\frac{\Delta x}{x}|Y|$$

form factor equation
$$f=\frac{\pi ab}{\left(2\pi \sqrt{\frac{a^2+b^2}{2}}\right)^2}$$

So the only part of this question I am having an issue with is the uncertainty part in part e). I have included the whole question as reference. So to derive the uncertainty in ff I used the uncertainty equation outlined above but the issue is that when I propagate the uncertainty I end up with an additional b term which I believe should cancel and I should end up with a numerical value, believe not 100% certain.

My working are as follows

$$\frac{\Delta f}{f}=2 \frac{\Delta P}{P}$$$$\frac{\Delta f}{f}=2 \frac{0.05}{2 \pi \sqrt{\frac{a^2+b^2}{2}}} $$

Now if i use the value give say a=b then if I sub that into my uncertainty equation I will be left with a b term, which is seems to confuse me as surely I would need a numerical value, as to me the question is asking for a numerical value, by asking for the uncertainty in f?

Now I am a bit ropy with uncertainty equations, but after a brief look at some examples, I can't see the equation I am using to be wrong, could anyone maybe advise me if I am going wrong as to why.

 

[mjc123]

You seem to be applying your uncertainty equation wrongly. If f = A/P, then n = -1 and
Δf = 1*(ΔP/P)*f
Δf/f = ΔP/P = 0.05
Your mistakes appear to be
(i) Where does the number 2 come from?
(ii) You are repeating f (or ff as you call it) - it should either be in the numerator of the RHS, or the denominator of the LHS, but not both.
(iii) 0.05 is ΔP/P, not ΔP, so don't divide by P.

 

[Taylor_1989]

Okay sorry i need to edit the OP the 2 comes from the nurmerator being square which I used a latex generator, I pasted the equation in incorrectly, my appoliges I will correct the.

 

[mjc123]

OK, you've corrected that, but you still wrongly have the expression for P in the denominator. 0.05 is ΔP/P, not ΔP. Do you understand what "5% uncertainty" means?