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.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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`




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

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial Team