Error Analysis: Solving 2a^2 + b^2 with Uncertainty (a=6.0+/-0.1, b=17.5+/-0.3)

[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!

 

[ice109]

whats the answer?

 

[AlephZero]

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?

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)

 

[ice109]

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)

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

 

[HallsofIvy]

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!

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".

 

[CRGreathouse]

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.

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.

 

[Diffy]

Exact value with 6 and 17.5 is 378.25