Using Differentials to find Error and Percent Error

grapeape

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!

Doc Al

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

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

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

I love this forum. Thanks for all the help!