Calculating the Uncertainty of a Pool's Volume

[jumbogala]

Homework Statement

Molly decides to find the volume of a swimming pool. She measures its length to 3 significant figures, its width to four significant figures, and its height to 2 signficant figures.

A) How many sig figs are in the volume of the pool?

B) How many sig figs should there be in the uncertainty of the volume of the pool?

Homework Equations

Volume = length x width x height

The Attempt at a Solution

A) The answer is 2, because it is the least amount of sig figs.

B) I don't know what an uncertainty is, or how to calculate it - maybe someone can explain it or give me a hint and I can try the problem again?

Edit: Maybe there's nothing to calculate - I just remembered there's a rule that says only 1 or 2 sig figs should be present in any error calculation.

 

[Haths]

Molly decides to find the volume of a swimming pool. She measures its length to 3 significant figures, its width to four significant figures, and its height to 2 significant figures.

A) How many sig figs are in the volume of the pool?

B) How many sig figs should there be in the uncertainty of the volume of the pool?

Lets say that the pool is 30.0m by 20.00m (L*W respectfully), now think about the nature of the measurements;

In the length:
3 lots of ten meters
0 lots of single meters
0 lots of ten centimeters

Take the last measurement, if we are only accurate to the nearest 10cm then we could be anywhere between +5 and -5 of a single ten centremetre graduation. In terms of meters that makes it +/-0.05. Where the '+/-' you should read 'plus minus' in reference to plus this amount to the value, or take this amount from the value.

This is because in rounding you round up on a .5 and round down on a .499999999... but as .4999999 is basically .5 we might as well call it .5

Thus we actually should write the measurements;

(30.0 +/- 0.05)m by (20.00 +/- 0.005)m

As you have correctly deduced that if the length can only be accurate to the nearest 10cm then there is little point knowing the width to the nearest 1cm because the first measurement wasn't that accurate.

So how many significant figures should we have in the uncertainty. Well witting the measurements with their margin of error should give you some idea.

Hint: Uncertainties are/never should be accurate...

Highlight the below if still stuck:

Well in the maximum case the length could be 30.5 and in the minimum case 29.5, the width 20.005 and 19.995

Therefore:
Max- 610.1525m²
Min- 589.8525m²

Now if we take the max away from the min;

= 20.3m²

We are out by an entire 20 metres! If we half this for our +/- we get 10.15m² in the strict sense this could be our uncertianty in the overal area messurement, but you might as well round it to 1 significant figure. This is because your length messurement is only to 2 significant figures and if I write the numbers like;

610.000000
589.000000
010.000000

You can see that the 10m is going to dominate the 15cm after the decimal point. Hence you should only quote the uncertainty to the first significant figure of your +/- uncertainty.

Hence (600 +/- 10)m

Note in practice, you don't need to go though the entire calculation, simply quoting a 1 significant figure above the lowest in your product will do, although try it for the volume and then see what result that shows...



You want the volume so you might want to do the method again with your third measurement.

You can't really be 'wrong' with uncertainties, because they are a judgment of how well you can measure something, and if your unsure, just dock a decimal figure or two from your answer.

Haths

 

[thomate1]

I can provide some guidance on how to approach this problem. The uncertainty in a measurement refers to the range of values within which the true value is likely to lie. In other words, it is an estimate of the precision or accuracy of a measurement. In this case, the uncertainty of the pool's volume would depend on the uncertainties in the measurements of its length, width, and height.

To calculate the uncertainty in the volume, you would need to consider the uncertainties in each measurement and how they affect the final result. The general rule is that the uncertainty in a calculated value should not exceed the precision of the least precise measurement used in the calculation.

In this case, the length has 3 significant figures, the width has 4 significant figures, and the height has 2 significant figures. To determine the uncertainty in the volume, you would need to consider the uncertainties in each measurement and use the least precise measurement to determine the final uncertainty. For example, if the uncertainty in the length measurement is ±0.1 cm, the uncertainty in the width measurement is ±0.001 m, and the uncertainty in the height measurement is ±0.01 m, then the final uncertainty in the volume would be ±0.1 m^3 (since the height measurement is the least precise).

In summary, to determine the uncertainty in the pool's volume, you would need to consider the uncertainties in each measurement and use the least precise measurement to determine the final uncertainty. This would result in a value with the appropriate number of significant figures, which in this case would be 1 or 2 significant figures. I hope this helps in your understanding of uncertainty and how to calculate it in a measurement.