# How Do You Calculate Uncertainty in the Volume of a Cookie?

• lep11
In summary, the cookies have an average volume of 2.8 cm3 and a 95% uncertainty in the volume. The ratio of the diameter to the thickness is 170 +/- 3% uncertain.
lep11

## Homework Statement

As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of 8.50 ± 0.02 cm and a thickness of 0.050 ± 0.005 cm. (a) Find the average volume of a cookie and the uncertainty in the volume. (b) Find the ratio of the diameter to the thickness and the uncertainty in this ratio.

## Homework Equations

V=pi*r2*h
d=8.50 +/- 0.02 cm → r=4.25+-0.01 cm

## The Attempt at a Solution

a.) average volume V=pi*r^2*h=pi*(4.25cm)2*0.050cm≈2.8cm3

volume of the smallest cookie V=pi*(4.24cm)2*0.045cm=2.54152...cm3
volume of the biggest cookie V=pi*(4.26cm)2*0.055cm=3.1356...cm3
uncertainty= (biggest - smallest)/2=0.30cm3
b.) ratio of the diameter to the thickness= 8.50cm/0.050cm=170
uncertainty=biggest ratio-smallest ratio=8.52cm/0.045cm-8.48cm/0.055cm=35
Please check my answers. I am not sure what is meant with that uncertainty thing.

Last edited:
The idea is that an actual value of a given quantity must lie between the biggest and smallest values and the assumption is made that it will most likely be right in the middle of that range. Half the spread should be "to the left" and half should be "to the right" of the average value. So, divide the total range by two to yield the uncertainty. In other words for f ±Δf you want:
$$Δf = \frac{f_{big} - f_{small}}{2}$$

In part b, check the thickness value you've used for the "biggest" term.

1 person
So now the volume could be expressed as V=2.8 +-0.30cm3 ?

lep11 said:
So now the volume could be expressed as V=2.8 +-0.30cm3 ?

V = 2.8 ± 0.3 cm3

Don't add a '0' past the digit you've rounded! That would imply another digit of accuracy, but you've already disposed of it by rounding.

gneill said:
In part b, check the thickness value you've used for the "biggest" term.

lep11 said:

You wrote:
b.) ratio of the diameter to the thickness= 8.50cm/0.050cm=170
uncertainty=biggest ratio-smallest ratio=8.52cm/0.45cm-8.48cm/0.055cm=3.569...
Shouldn't that be 0.045 cm?

gneill said:
You wrote:

Shouldn't that be 0.045 cm?
Yes it should! A typo. Thanks. Is b otherwise correct or should it be 35/2?

Last edited:
lep11 said:
Yes it should! A typo. Thanks. Is b otherwise correct?

Not correct. Redo the calculation and remember what I said about dividing the result in half.

1 person
I appreciate your help. However, I still don't know what was the whole exercise for.

Last edited:
lep11 said:
I appreciate your help. However, I still don't know what was the whole exercise for.

The idea of uncertainty values is to establish a range over which the results of calculations may lie when they are derived from imperfect data. Every measurement has an error associated with it -- no measurement is ever perfectly accurate.

You will use uncertainty calculations a LOT in the sciences, particularly if you have to do labs and the associated lab reports.

## 1. What is uncertainty in the volume?

Uncertainty in the volume refers to the lack of exactness or precision in the measurement of the volume of a substance or object. It is the range of possible values within which the true volume is likely to fall.

## 2. What factors contribute to uncertainty in volume measurements?

Several factors can contribute to uncertainty in volume measurements, including the precision of the measuring instrument, the skill of the person taking the measurement, and the variability of the substance being measured.

## 3. How is uncertainty in volume expressed?

Uncertainty in volume is typically expressed as a percentage or a range of values. For example, a volume measurement of 25 mL with an uncertainty of ±2% means that the true volume is likely to be between 24.5 mL and 25.5 mL.

## 4. Can uncertainty in volume be reduced?

Yes, uncertainty in volume can be reduced by using more precise measuring instruments, improving measurement techniques, and reducing the variability of the substance being measured. However, it is impossible to completely eliminate uncertainty.

## 5. How does uncertainty in volume affect scientific experiments and research?

Uncertainty in volume can have a significant impact on the accuracy and reliability of scientific experiments and research. It is important for scientists to understand and account for uncertainty in volume in order to draw accurate conclusions and make informed decisions based on their findings.

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