Given that A=2, B=4 and C=3, both rounded to the nearest integer. Calc the absolute & the relative error for:?

Question

a) √7(B+C)

b) ABC - pie, pie = 3.14159

Answer 1

Assuming that:
1. The errors for the three variables are uniformly distributed over their possible values of half a unit on either side of nominal,

and especially 2. The errors are not correlated with each other, e.g., no systematic errors would tend make all the errors pile up on the same side of the nominal value,

then:
1. The statistical variance for each datum is 1/12 and its standard deviation is √3/6.

2. If the values are added or subtracted, the variances will add.

3. If the values are scaled by a constant, the variances will scale by the square of that constant.

4. Adding or subtracting a constant leaves the variances and standard deviations unchanged.

5. If the values are multiplied or divided together, their relative variances - squares of the quotient of the standard deviations divided by the nominal values - will add. This one is somewhat less well-justified in theory because the nonlinearity of the multiply and divide operations will cause the distributions to skew a bit, particularly when the relative standard deviations are comparatively large as they are especially for A.

For a) The standard deviations for B and C will be √3/6, which will each give 1/12 when squared. Adding these variances gives 1/6 for the variance of the sum. Multiplying this by the square of the multiplying constant √7 gives a variance of 7/6 and a corresponding standard deviation of √42/6 for the overall expression. The relative standard deviation will be this value, divided by the nominal value of √7(4 + 3), for a relative standard deviation of √6/42.

b) The relative variance for A is 1/(12*A²) or 1/48; for B, 1/192; and for C, 1/108. The sum of these relative variances is 61/1728. The (absolute) variance for the subexpression ABC is the relative variance multiplied by the square of the nominal value of 24 for ABC, giving an absolute variance of 61/3, which will not be changed by subtracting the constant pi. The standard deviation for this subexpression is therefore
√183/3. The relative standard deviation will be this quantity divided by the nominal value for the expression of 24 - π to give a final value for the relative standard deviation of √183 / (72 - 3π).

Answer 2

a)
√7(B+C)
=√7(4+3)
=√7x7
=7

b)
ABC - pie
=2 x 4 x 3 - 3.14159
=8 x 3 - 3.14159
=24 - 3.14159
= 20.85841

Answer 3

pie....yummy....uhh what sorry your question gave me a headache..got tylenol :)