The measurement of the circumference of a circle is found to be 64 centimeters, with a possible error of 0.9 centimeter. Approximate the percent error in computing the area of the circle.

MY ANSWER 8.83%

C = 2 pi r = 64 {circumference is 2 pi r}

so r = 64/(2pi) = 10.19

dC/dr = 2 pi { just taking the derivative of 2 pi r}

so dC = 2 pi dr {multiplied both sides by r}

.9 = 2 pi dr (used .64 by mistake) {the .9 cm was given in the problem statement for dC}
so
dr = .1433 {divided by 2 pi}

A = pi r^2 = pi (10.19)^2 = 325.95
{because area = pi r^2 of circle}

dA/dr = 2 pi r (of course:) { taking derivative but if we sketch we see that increase in area is circumference times dr}
then I inverted some lines

dA = 2 pi r dr {multiplying both sides of above by dr
dA = 2 pi (10.19)(.1433) { putting in the numbers for r and dr }
dA = 9.17

percent error = (9.17/325.95)100 = 2.81%
{because we want dA/A * 100}

I made that a lot harder than it needed to be but follow it and you will be able to shorten it.

Also, it says to estimate the maximum allowable percent error in measuring the circumference if the error in computing the area cannot exceed 3%. How do I do this?

part 2

(dA/A)100 = 3

(2 pi r dr /pi r^2)100 = 3
or
dr/r = .015
but dr = dC/(2pi)
so
dC/2 pi r = .015
or
dC/C = .015
so 1.5 % error in C gives 3% error in A

Okay, I really don't understand how you did any of that. What's the point of half of it? I tried to do some of what you did, but I got different numbers for my answers.

well, let me do the second part slowly

C = 2 pi r
dC = 2 pi dr
so
dC/(2 pi r) = dr/r
or
dC/C = dr/r which any architect will tell you

dA = C dr
dA/A = C dr/A
dA/A = C dr/(pir^2)
dA/A = 2 pi r dr/(pi r^2)
dA/A = (2)dr/r
SO
dr/r = (1/2) dA/A Important
if dA/A = 3%
then dr/r = half that or 1.5 %

C = 2 pi r = 64

so r = 64/(2pi) = 10.19
dC/dr = 2 pi
so dC = 2 pi dr
.9 = 2 pi dr (used .64 by mistake)
so
dr = .1433

A = pi r^2 = pi (10.19)^2 = 325.95
dA/dr = 2 pi r (of course:)
percent error = (9.17/325.95)100 = 2.81%
dA = 2 pi r dr
dA = 2 pi (10.19)(.1433)
dA = 9.17

note that if I had done the second part first I would have known that

dA/A = 2 (dC/C)
dA/A = 2(.9/64)
= .028
which is 2.8 percent which I got after much algebra

Honestly, I just waited till he explained it on board. i just put random stuff. :)

C = 2 pi r = 64

so r = 64/(2pi) = 10.19
dC/dr = 2 pi
so dC = 2 pi dr
.64 = 2 pi dr
so
dr = .10186

A = pi r^2 = pi (10.19)^2 = 325.95
dA/dr = 2 pi r (of course:)

dA = 2 pi r dr
dA = 2 pi (10.19)(.10186)
dA = 6.52

percent error = (6.52/325.95)100 = 2%