# A window has the shape of square surmounted by a semicircle. Window base has width 60cm with a possible error of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of the window. Given a window that is in the shape of a square of side 60cm surmounted by a semicircle, estimate the error in computing the area if there is a possible error of 0.1cm in determining the side length.

(1) The propagated error can be found by `f(x+Deltax)-f(x)=Deltay` where `f(x+Deltax)` is the actual value, `Deltax` the possible error, `f(x)` the measured value, and `Deltay` the propagated error.

(2) We can estimate `Deltay` by `dy` ; `Deltay=f(c+Deltax)-f(c)~~f'(c)Deltax=dy`

(3) Let the side length of the square be `s` ; then the radius of the semicircle is `1/2 s` . The area of the window is `A=s^2+1/2(pi (1/2 s)^2)` or `A=s^2+(pis^2)/8` .

Then `(dA)/(ds)=(2s+(pis)/4)`

(4) Now:

`Delta A~~dA`

`=(2s+(pis)/4)ds`

`=(2(60)+(pi(60))/4)(0.1)`

`~~16.71cm^2`

So the potential maximum error in computing the area of the window is approximately `+- 16.71cm^2`