# Error Analysis Question

1. Sep 20, 2007

### neoking77

given that a=(6.0+/-0.1) and b=(17.5+/-0.3)

what is 2a^2 + b^2

i'm having so much trouble with this
i think we first go (6.0+/-0.1)^2 then times it by 2, and add (17.5+/-0.3)^2...i tried this but couldn't get the right answer...what am i doing wrong?
...ANY help would be greatly appreciated...i feel if someone will help me with just this question i can figure out how to do the rest. thank you in advance!!

2. Sep 20, 2007

### ice109

3. Sep 20, 2007

### AlephZero

That seems like the right idea.

If you show us your arithmetic and tell us what the "right" answer is supposed to be, somebody can probably help you more.

I wonder if you wre supposed to use the approximate formula

(x +/- y)^2 = (x^2) +/- (2xy)

not the exact formula

(x +/- y)2 = (x^2+y^2) +/- (2xy)

4. Sep 20, 2007

### ice109

when multiplying or adding quantities with small uncertainties, add uncertainties. i think.

5. Sep 20, 2007

### HallsofIvy

Staff Emeritus
I'm not sure what you mean by (6.0+/-0.1)^2. I presume you mean that you make two calculations: The largest that 6.0+/- 0.1 can be is, of course, 6.1. 6.12= 37.21 and 2 times that is 74.42. The largest that 17.5+/-0.3 can be is 17.8. 17.82= 316.84. Since we are adding positive numbers (you have to be careful about that) the largest possible value of 2a2+ b2 is 391.26.

Similarly, the smallest possible value of 6.0+/-0.1 is 5.9. Using that value for a, 2a2= 2(5.9)2 is 69.62. The smallest possible value of 17.5+/-0.3 is 17.2. Its square is 295.84. The smallest possible value of 2a2+ b2 is the sum of those: 356.46.

That is, if a= 6.0+/-0.1 and b= 17.5+/-0.3, then 2a2+ b2 can lie any where between 356.46 and 391.26. You will notice that the value 2(62)+ 17.52= 378.25 is between those but not exactly in the middle: the midpoint of the interval is (391.26+ 356.46)/2= 373.86.

To write that in the form x+/- d, we can do either of two things:
(1)take the midpoint of the interval, 373.86 and, since 373.86- 356.46= 391.20-373.86= 17.4, the value is 373.86+/-17.4 or
(2)take the value given by 6.0 and 17.5, 376.25 and noting that 391.26- 376.25= 15.01 and 376.25-356.46= 19.79 (the interval is not symmetric about that value), use the larger of those, 19.79, to be sure we are inside the interval: 376.25+/-19.79.
The latter is typically easier to calculate but allows for a larger "error" than necessary.

We could also approximate as follows: if f= 2a2+ b2, then, differentiating, df= 4ada+ 2bdb. "df", the differential, is not exactly the "error" but if the error is small, they are close. Here, df= 4(6.0)(0.1)+ 2(17.5)(0.3)= 12.9 giving 376.25+/- 12.9 as the estimate.

There is an engineering "rule of thumb" that "When quantities add, the errors add. When quantities multiply, the relative errors add."
Specifically, d(a+ b)= da+ db so the "errors add"
d(ab)= bda+ adb and, dividing by ab, d(ab)/ab= da/a+ db/b. da/a and db/b are, of course, the "relative errors".

6. Sep 20, 2007

### CRGreathouse

It's 365.46; you reversed the digits. I'll choose to let neoking77 fix any errors that have propagated with that one -- it's his problem, after all.

7. Sep 20, 2007