Suppose a cone has a radius, r, of 15.63 mm measured by the metric micrometer sh

Question

Suppose a cone has a radius, r, of 15.63 mm measured by the metric micrometer shown below. The distance s (see below) is equal to 3.5 times r. a) Calculate the surface area and volume of the cone. b) calculate the uncertainty in the surface area and volume. c) calculate the percent uncertainty in volume and surface area. Recall that Vcone= (1/3)*pi*r^2*h and Acone= pi*r*s + pi*r^2. Note that the uncertainty of the instrument is 10% of the smallest scale division and that the major divisions on the micrometer are millimeters.

Explanation / Answer

part a:

surface area=pi*r*s+pi*r^2

=3.4537*10^(-3) m^2

from the right angle formed by the h,r and s:

h=sqrt(s^2-r^2)=0.05242 m

volume=(1/3)*pi*r^2*h

=(1/3)*pi*0.01563^2*0.05242=1.341*10^(-5) m^3

part b:

uncertainity in surface area=uncertainty in radius+uncertianty in s

=2*uncertainty in the instrument

=2*0.1*0.001

=2*10^(-4)

hence value of uncertainty=2*10^(-4)*surface area=6.9074*10^(-7) m^2

uncertainty in volume=2*uncertainty in radius+ uncertainty in height

=3*uncertainty of the instrument

=3*0.1*0.001

=3*10^(-4)

value of uncertainty in volume=4.023*10^(-9) m^3
part c:

percentage uncertainty in surface area=2*10^(-4)*100=0.02%

percentage uncertainty in volume=0.03%

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