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Using Differentials to find Error and Percent Error

by grapeape
Tags: differentials, error, percent
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grapeape
#1
Oct20-07, 04:58 PM
P: 10
1. The problem statement, all variables and given/known data
One side of a right triangle is known to be 20 cm long and the opposite angle is measured as 30(degrees), with a possible error of +/- 1 degree.
a) Use differentials to estimate the error in computing the length of the hypotenuse
b) what is the percentage error.


2. Relevant equations



3. The attempt at a solution
Well, using the given data I found that the hypotenuse when x=30(degrees) is 40 cm. The equation I used was h(x)=20/sin(x). I know that the change in (h) is equal to error x h'(x). When finding h'(x) I got -20cos(x)/sin(x)^2. I'm not sure if this is correct. My book doesn't do a great job at explaining anything so any help will be greatly appreciated!
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Doc Al
#2
Oct20-07, 05:37 PM
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You're almost there. Expressed in terms of differentials, h' = dh/dx. So write dh in terms of dx. dx will represent the error in angle (be sure to express dx in radians, not degrees); the equation for dh will tell you the corresponding error in the hypotenuse.
grapeape
#3
Oct20-07, 08:12 PM
P: 10
so do I do ( -20cos(30)/sin(30)^2 ) x pi/180? if so I got -1.209. Can error be negative? Or since it's both +/- 1 degree, do you also do another equation multiplying by a negative pi/180. My real questions are, how is error represented (multiple numbers, one number, a continuum)? And, how do I go about converting this to a relative error, and then percent error.

Thanks!

Doc Al
#4
Oct21-07, 06:05 AM
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Using Differentials to find Error and Percent Error

Quote Quote by grapeape View Post
so do I do ( -20cos(30)/sin(30)^2 ) x pi/180? if so I got -1.209.
Looks good. The angle is 30 +/- 1 degrees, so the hypotenuse is 40 +/- 1.2 cm. You can express that as a percent error--what percent of 40 is 1.2?
grapeape
#5
Oct21-07, 06:08 PM
P: 10
I love this forum. Thanks for all the help!


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