The volume of a cylinder is given by the expression
V=πr2h
The uncertainties for V and h are as shown below
V±7%h±3%
What is the uncertainty in r?
Now, the obvious answer would be 2%, from dVV=dhh+2drr
However, rearranging to r2=Vπh gives 2drr=dVV+dhh
How do you explain this?
(The mark scheme lists 5% as the correct answer)
Answer
You're confusing independent and dependent variables. When you propogate from uncertainties in the xi to some f(x1,x2...), the formula δf(x1...)=∑|∂f∂xi|δxi assumes that each of the xi is an independently measured variable and that f is a dependent variable to be calculated from the xi.
In the example you give, you have two independent measurements of V and h and are expected to calculate the uncertainty in r. Well, to use the above formula, you need to write r as a dependent variable of V and h. Therefore, it's only correct to solve for r first, and then calculate the uncertainty.
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