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I'm stuck with the following problem:

Estimate the relative uncertainty of the surface of a sphere if the measure of its radius is accurate to $2.0 \%$.

My work so far:

Sphere surface: $ 4\pi r^2 $

Formula for relative uncertainty: $ \left(\dfrac{|df|}{f} \times 100 \right) \% $ is equal to $ \left(\dfrac{\left \lvert f'(x)dx \right \rvert}{f(x)} \times 100 \right) \% $

My function : $ F(x) = 4\pi r^2 $

Thanks.

2 Answers 2

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Your function might be $F(x)=4\pi x^2$, or it might be $F(r)=4\pi r^2$, but it's certainly not $F(x)=4\pi r^2$. Now that we have that out of the way, can you see how to use all those formulas you have?

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    If you put that missing $r$ in the numerator, you will get something involving $100dr/r$, and that's the relative uncertainty in $r$, and the problem statement tells you the relative uncertainty in $r$ is 2%, so you'll have your answer.2012-03-13
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Hint: I can take issue with the way your formula is expressed, but the basic idea is correct. I would write it as $\frac {\Delta f(x)}{f(x)} \approx \frac 1{f(x)}\frac {df(x)}{dx} \Delta x$ which shows clearly that you are progressing along the tangent to the curve and this is acceptable as long as the higher derivatives don't cause a problem. So first, you need to find $\frac {dF(r)}{dr}$ (and please be careful with capital letters-$f$ is not the same as $F$).

The basic idea comes from $(1+x)^n \approx 1+nx$ if $x \ll 1$ so something that goes at the $n^{th}$ power moves $n$ times as fast at the base.