We measure the radius r of a circle and obtain that that r=9 centimeters. We use this value to compute the area of the circle. We know that the error in our measurement is at most 1.6%. Use the linear approximation to the area A(r)=pi*r^2, at r=9 to estimate the maximum error in our computed value for the area.
I get (162/125) * pi as my answer. :/
Is the equation not 2 * pi * 9 * .016?
A slightly different way of looking at this. so the derivative is . At r= 9, and [tex]A'(9)= 18\pi[itex]. The tangent line at r= 9, [tex]18\pi(r- 9)+ 81\pi[/itex] gives the linear approximation. The "change" is . 1.6% of 9 is 0.144 so evaluate for r= 9.144.
There is, by the way, an engineers "rule of thumb" that is based on this: When adding (or subtracting) quantities add their errors. When multiplying (or dividing) add their percentage errors. Here we are multiplying r times r. since the percentage error for r is 1.4%, the percentage error for , and so for would be 1.4+ 1.4= 2.8%