Question

About how accurately must the interior diameter of a 10-m high cylindrical storage tank be measured to calculate the tank's volume within 1% of its true value?

About how accurately must the tank's exterior diamter be measured to calculate the amount of paint it will take to paint the side of the tank to within 5% of the true amount?

Answer 1

The formula for cylinder volume is
V = π•r²•h
Let's suppose the radius is 5 meters
volume = 785.398
For 5.1
volume = 817.128
So a change of .1 meters in the radius causes a 4.04% increase in volume

If radius = 5
volume = 785.398
radius = 5.02
volume = 791.6939150752 a .8 % difference

radius = 5.03
volume = 794.851 a 1.2 % increase

Answer 2

So for a 5 meter radius, it must be measured with an accuracy of .02 meters (or 2 centimeters) to have the accuracy within 1% of the volume.

Answer 3

V = (pi) * R^2 * H
dV = 2 * (pi) * R * H * (dR)
dV/V = 2 * (pi) * R * H * (dR) / (pi) * R^2 * H = 2 * (dR) / R
2 * (dR) / R = dV/V
dR/R = 1/2 * dV/V
dV/V * 100 is the percentage error in volume calculation = 1%
dV/V = 1/100 = .01
Therefore, dR/R = 1/2 * dV/V = 1/2 * .01 = .005 or 0.5%

The percentage error in measuring the radius (or the diameter) must be within 0.5% of the true radius (or the diameter) in order to calculate the tank's volume within 1% of its true value.

Answer 4

In Wolf1728's example above with radius = 5 m,
the measurement should be within 5 * .5 / 100 = 0.025 m in order for the volume to be within 0.1% error as is confirmed by his calculations.

Answer 5

Well it's good to see somebody verified my thoughts on the subject.
Anyway ranga, I like your explanation better than mine.

Answer 6

Thanks wolf1728.

Answer 7

u r welcome