The area of a circle is to be computed by measuring the radius. the actual radius is r=2.67, but it was measured with an error. it was measured to be r= 2.69
a) find the exact error in the radius
b) find the exact error in the area
c) find the approximate error in the area
d) find the relative error in the radius and the percent error in the area using your answer from part (b)

A) The exact area is just the difference between the two numbers:
     2.69-2.67 = 0.02 units

b) Calculate the area of the circle with the two values for the radius:
     A = πr^2 = π * 2.69^2 = 22.7328786 units squared
     A = πr^2 = π * 2.67^2 = 22.39609987 units squared
     Then, find the difference between the two numbers:
     22.7328786 - 22.39609987 = 0.336778731 ≈ 0.34 units squared (2.s.f.)

c) I'm not sure what the question is asking for (perhaps the rounding expressed      at the end of b?)

d) The relative error is the absolute (exact) error divided by the actual value for         the radius:
    0.02/2.67 = 0.007490636704 ≈ 0.0075 (2.s.f.)

    The percentage area is the absolute (exact) error divided by the actual value     for the area, and then multiplied by 100 to find a percentage:
    (0.336778731/22.39609987)*100 = 1.503738298 ≈ 1.50% (2.s.f.)
         

I'm not sure about this question ... Some of the question could have been
written better.  Idon't know ... maybe all the terms were clearly defined in this class, and my math is just too old.  But here's how I see it:

a).
The question says that the actual real radius is 2.67, but whoever measured it was careless and said that it's 2.69. 
The exact error is (2.69 - 2.67) = 0.02 .

===>  The area of any circle is (pi) (r-squared).  (You need to know this.)

b).
The actual real area is (pi) (2.67-squared) but he said that the area is
(pi) (2.69-squared). 
The exact error in the area is (pi) (2.69-squared minus 2.67-squared).
We can never say exactly what (pi) is, but we can get the exact value of
(2.69² - 2.67²), because we know that (a² - b²) factors into (a+b) (a-b).

(2.69² - 2.67²) = (2.69 + 2.67) (2.69 - 2.67) = (5.36) (0.02) = 0.1072 .

So the exact error in the area is (pi) (0.1072) .

c). 
When you plug in some number for (pi), you get an approximate number for the actual error in the area.   The more complicated your number for (pi) is, the more accurate the answer will be.  But since (pi) can never be exactly written, the number will never be perfectly exact, and will always be an approximation.

Let's use ... I don't know ... let's use, say, 3.142 for (pi).

Then the approximate error in the area is (3.142) (0.1072) =  0.3368 .

The better your (pi) is, the better your approximation is.  But it can never be exact.

d).
I'm not sure what the "relative error" in the radius means.
The error in the radius is (0.02) out of (2.67).  That's about 0.75% too big.

Now, they want us to find the percent error in the area using our answer
to part b).  This could be tricky.

Oh !  I see how to do it now . . .

-- We know that the real actual area is   (pi) (2.67-squared) . 

-- In part b), we found that the exact error in the area is (pi) (0.1072).

-- So we just have to find what percent (pi) (0.1072) is,
out of  (pi) (2.67-squared), and that's the percent error.  

To find a percent, we divide the little piece by the whole thing,
then multiply the quotient by 100.

(100) (pi) (0.1072) / (pi) (2.67-squared)  =  10.72 / (2.67-squared) = 1.5 %

NOTICE something ... The percent error in the area (1.5%) is double the percent error in the radius (0.75%).  I think that happens because the radius gets squared in order to find the area, so the error in the radius gets applied twice.

And thank you for your generous 5 points !  The mouldy crusts and grey cloudy water are delicious.