I really don't understand, can we measure the circumference of the sphere?
what is the formula?
Hello , please show me how to solve this problem.
The circumference of a sphere was measured to be 84cm with a possible error of 0.5
a. Use differentials to estimate the maximum error in the calculated surface area. What is the relative error?
b. Do the same thing calculate volume.
C = 2pi*r
=> dC = 2pi dr
=> 0.5 = 2pi dr
=> dr = 1/4pi
now with the error:
C = 2pi*r
=> r = C/2pi = 84/2pi = 42/pi
Now A = 4pi*r^2
=> dA = 8pi*r dr
=> dA = 8pi*(42/pi)(1/4pi)
=> dA = 336/pi ...........the maximum error in surface area.
to find the relative error, we divide the error in area by the total area.
dA/A = 8pi*r dr/4pi*r^2 = 2 dr/r = 2[(1/4pi)/(42/pi)] = 2(1/168) = 0.0119 which is about 1.2% relative error
we have r = 42/pi and dr = 1/4pi
now V = (4/3)pi*r^3
=> dV = 4pi*r^2 dr
=> dV = 4pi*(42/pi)^2 * (1/4pi)
=> dV = 1764/pi^2 = 178.73 approx ...maximum error in calculated volume
So relative error is given by
dV/V = (4pi*r^2)/(4/3)pi*r^3 = 3 dr/r = 3[(1/4pi)/(42/pi)] = 3(1/168) = 0.018 or about 1.8 % relative error