# Error calculation

1. Sep 27, 2013

### steve2510

1. The problem statement, all variables and given/known data
Two Sides of a triangular plate are measured as 125mm and 160mm, each to the nearest millimetre. The included angle is quoted as 60+-1. Calculate the length of the remaining side and the maximum possible error in this result

My first question is to the nearest millimetre? What does this mean, i think +-0.5 but when i think about this if it was 0.5 biggest it would be rounded up..

2. Relevant equations
δu = Fxδx + Fyδy + Fzδz

3. The attempt at a solution
So firstly to find the length of the other side i used the cosine rule
a^2 = b^2 + c^2 - 2bccos(A)
Where b = 125
c = 160
A = pi/3
Which comes out as 145.7 which is correct.
The next stage i guess is to take partial derivatives.
My equation for a is a^2 so i assume i have to square root it so i can find δa im not sure but i guess i have to??
δa = partial with respect to b * δb + partial with respect to c *δc + partial with respect to A δA

Therefore a = (b^2 + c^2 - 2bcCos(A))^1/2
Okay this is where it gets messy.
Partial with respect to b:
[1/2 * 2b-2cCosA/(b^2+c^2-2bcCos(A))^1/2] *δb

Partial with respect to C
[1/2 * 2c - 2bcos(A)/((b^2+c^2-2bcCos(A))^1/2) ]*δc

Partial with respect to A
[2bcsinA/((b^2+c^2-2bcCos(A))^1/2)] *δA

When i input values of δb and δc = +-0.5 and δA = +-1 i don't seem to get the right answer which is +- 2.6, i really don't see where I'm going wrong, as i did the similar method with a previous exercise.

2. Sep 27, 2013

### Staff: Mentor

Right.
True, but 0.4999999 (add as many finite 9 as you like) would be rounded down, the smallest upper bound is 0.5.

You can calculate δa^2 as intermediate result if you like.

I think there are some brackets missing in your derivatives.

Last edited: Sep 28, 2013
3. Sep 27, 2013

### steve2510

Partial b = [((2b-2cCosA)/2*(b^2+c^2-2bcCos(A))^1/2] *δb
= [((250 - 160)/2*((125^2 + 160^2 - 2*160*125*0.5)^0.5))] * 0.5
= (90/291.4) * 0.5 = +-0.154
partial c has the same denominator
[((2c - 2bcos(A)) / 291.4) ]*0.5
= 195/291.4 * 0.5 = +-0.335
Partial A has the same denominator aswell
= (125*250*2*sin(pi/3)) / 291.4
Which is a really large number, not sure whats going wrong there..

The correct answer is 2.6, i don't seem to be close to that at present, i guess there must be a simple mistake i'm making.

4. Sep 27, 2013

### vela

Staff Emeritus
Did you convert $\delta A$ to radians as well?

5. Sep 27, 2013

### steve2510

Nope completely forgot about that, thanks
Well that leaves me with
[(125*250*2*sin(pi/3) / 291.4 ] *pi/180 = 3.24 which is still off, i cant see where i've made a mistake, i've done problems after this one and been fine, i just can't seem to get this one.

6. Sep 27, 2013

### vela

Staff Emeritus
The 250 in the numerator should be 160, no?

Also, I get a slightly different expression for the partial derivative wrt A than you do.

7. Sep 28, 2013

### steve2510

δa = $\frac{\partial a}{\partial b}$ δb + $\frac{\partial a}{\partial c}$ δc + $\frac{\partial a}{\partial A}$ *δA

δb = 0.5 δc = 0.5 δA = pi/180

$\frac{\partial a}{\partial b}$ = $\frac{b-cCos(A)}{\sqrt{b^{2}+c^{2} - 2bccos(A)}}$
$\frac{\partial a}{\partial b}$ = $\frac{125-160*Cos(\frac{pi}{3})}{\sqrt{125^{2}+160^{2} - 2bccos(\frac{pi}{3})}}$ = $\frac{45}{145.688}$
$\frac{\partial a}{\partial b}$ = 0.309
$\frac{\partial a}{\partial b}$ * δb = 0.1545

$\frac{\partial a}{\partial c}$ = $\frac{c-bCos(A)}{\sqrt{b^{2}+c^{2} - 2bccos(A)}}$
$\frac{\partial a}{\partial c}$ = $\frac{160-125*Cos(\frac{pi}{3})}{\sqrt{125^{2}+160^{2} - 2bccos(\frac{pi}{3})}}$ = $\frac{97.5}{145.688}$
$\frac{\partial a}{\partial c}$ = 0.669
$\frac{\partial a}{\partial c}$ * δc = 0.335

$\frac{\partial a}{\partial A}$ = $\frac{cbsin(A)}{\sqrt{b^{2}+c^{2} - 2bccos(A)}}$
$\frac{\partial a}{\partial A}$ = $\frac{160*125*sin(\frac{pi}{3})}{\sqrt{125^{2}+160^{2} - 2bccos(\frac{pi}{3})}}$ = $\frac{365.52}{145.688}$
$\frac{\partial a}{\partial A}$ = 2.509
$\frac{\partial a}{\partial A}$ * δA = 0.044

I know for a fact these won't add up to 2.6, so i very confused at which stage im going wrong.

8. Sep 28, 2013

### Staff: Mentor

160*125*sin(pi/3) is not 365.52. Make sure that you use radians for the sine, not degrees.

9. Sep 28, 2013

### steve2510

oo what a silly mistake
$\frac{\partial a}{\partial A}$ = $\frac{160*125*sin(\frac{pi}{3})}{\sqrt{125^{2}+160^{2} - 2bccos(\frac{pi}{3})}}$ = $\frac{17320.51}{145.688}$
$\frac{\partial a}{\partial A}$ = 118.888
$\frac{\partial a}{\partial A}$ * δA = 2.075

2.075 + 0.1545 + 0.335 = δa
δa = 2.56 → 2.6 which is the correct answer yey, thanks very much.