Given that V=[ð*D^2]*h/4
and
fractional error in W leading from the errors Äa, Äb and Äc from the measurement of a, b, and c is given by
ÄW/W= sqrt((nÄa/2)^2+(mÄb/2)^2+(pÄc/2)^2)
where one has a quantity W proportional to a^n* b^m* c^p,where a, b, and c are variables
calculate the ratio of the error in the volume to the mean volume (ÄV/V) in terms of the corresponding errors in the measurement of the diameter (ÄD) and the height (Äh)
If you use ISO-8859-1 encoding to send your text, it would be a little easier for us to understand your post.
As it is, many symbols are not obvious, although some are understandable.
(Ä)=δ
(ð)=π
Also, without a complete description of your problem, it is difficult to figure out what you need.
V=πD²h/4 is the volume of a cylinder. But it is not obvious where do the symbols W,m,n,p and a,b,c come from.
With MathMate's descriptions of symbols,
ln V = ln(pi/4) + 2ln D + lnh
Now take the differential of both sides.
äV/V = 2 äD/D + äh/h
If the measurement errors in D and h are random and uncorrelated, the rms error in the measured volume is
(äV/V)rms = sqrt[4(äD/D)^2 + (äh/h
)^2]
according to the root-sum-of-squares (RSS) rule.
To calculate the ratio of the error in the volume (ÄV/V) in terms of the errors in the measurement of the diameter (ÄD) and the height (Äh), we need to find the fractional errors in the diameter and height and substitute them into the given formula.
First, let's find the fractional error in the diameter (ÄD). Since the volume formula V depends on the diameter squared, we can use the chain rule to find the fractional error:
ÄD/D = 2 * ÄD/D
Next, let's find the fractional error in the height (Äh):
Äh/h = Äh/h
Now, let's substitute these fractional errors into the formula for the fractional error in W:
ÄW/W = sqrt((n * Äa/2)^2 + (m * Äb/2)^2 + (p * Äc/2)^2)
Since V is proportional to a^1 * b^0 * c^0 = a, our W value will be W = V.
Substituting the fractional errors, we have:
ΔV/V = sqrt((1 * Δa/2)^2 + (0 * Δb/2)^2 + (0 * Δc/2)^2)
Simplifying further, we have:
ΔV/V = sqrt((Δa/2)^2)
Taking the square root of the squared term, we have:
ΔV/V = Δa/2
Therefore, the ratio of the error in the volume to the mean volume (ÄV/V) in terms of the corresponding errors in the measurement of the diameter (ÄD) and the height (Äh) is:
ΔV/V = ΔD/2
Note that the error in the height (Äh) does not contribute to the error in the volume (ÄV) since it is not present in the formula V = (π * D^2 * h)/4.