# Uncertainity and Error Question

• inno87
In summary, the conversation discusses a cylindrical cookie with uncertain dimensions, and the calculation of its volume and associated uncertainties. Specifically, it addresses the most likely value of the volume, the percent and absolute uncertainties, and the necessary reduction in the diameter's uncertainty to achieve a 3% uncertainty in the volume.
inno87
I'm not sure if I am doing this right, can anyone help me out?

A cylindrical cookie has a diameter of 5.0 $$\pm$$ 0.1 cm, and a thickness of 1.00 $$\pm$$ 0.01 cm.

A. Assuming the uncertainities are normally distrubited, what is the most likely value of the volume of the cookie?

V=pi*(d/2)^2*h=pi*(1 $$\pm$$ .01 cm * [(5.0 $$\pm$$ .1 cm)/2]^2=pi*[1 $$\pm$$ .01 cm * (2.5 $$\pm$$ .05 cm)^2=pi*[1 $$\pm$$ .01 cm * 6.25 $$\pm$$ .1 cm^2]=pi* 6.25 $$\pm$$ .11 cm^3 = 19.634 $$\pm$$ .11 cm^3

Most likely volume is 19.634 cm^3

B. What is the percent uncertainty in the volume?
.11/19.634=.5603%

C. What is the absolute uncertainty in the volume?
.11 cm^3 (taken from question 1)

D. Assuming the thickness uncertainity remains $$\pm$$ .01 cm, to what value would the diameter's uncertainty (in cm) have to be reduced in order to make the uncertainty in the volume $$\pm$$ 3%?

I tried setting up something like (.01+D_unc)/(19.634)=.03 but that would mean you'd have to increase the uncertainty so it seems I may have done something wrong here!

First of all, it is important to note that uncertainty and error are inherent in any measurement and calculation, and it is important to acknowledge and account for them in scientific work. In this case, the uncertainty in the diameter and thickness of the cookie will affect the calculated volume, and it is important to understand the impact of this uncertainty on the final result.

In order to calculate the most likely value of the volume of the cookie, we can use the formula for the volume of a cylinder. However, because both the diameter and thickness have uncertainties associated with them, we need to use the uncertainty propagation formula to calculate the overall uncertainty in the volume. This formula takes into account the uncertainties in each variable and calculates the overall uncertainty in the final result.

In this case, the most likely volume of the cookie is 19.634 cm^3, and the uncertainty in this value is 0.11 cm^3. This means that the actual volume of the cookie could be anywhere between 19.524 cm^3 and 19.744 cm^3, with a 68% confidence level. This is because the uncertainties in the diameter and thickness of the cookie contribute to the overall uncertainty in the volume.

The percent uncertainty in the volume is calculated by dividing the absolute uncertainty (0.11 cm^3) by the most likely value of the volume (19.634 cm^3) and multiplying by 100. This gives a percent uncertainty of 0.56%.

In order to reduce the uncertainty in the volume to ±3%, we need to reduce the overall uncertainty to 3% of the most likely value of the volume. This can be achieved by reducing the uncertainty in the diameter, as the thickness uncertainty is already at ±0.01 cm. Using the uncertainty propagation formula, we can calculate that the uncertainty in the diameter would need to be reduced to ±0.005 cm in order to achieve an overall uncertainty of ±3% in the volume.

In conclusion, uncertainty and error are important considerations in scientific work, and it is essential to understand their impact on calculations and results. By using appropriate formulas and techniques, we can accurately account for uncertainty and ensure the validity of our scientific findings.

## 1. What is uncertainty in scientific measurements?

Uncertainty refers to the potential for error or variation in a scientific measurement. It is a measure of the range of values that could reasonably be attributed to a particular measurement, taking into account any potential sources of error.

## 2. How do scientists account for uncertainty in their results?

Scientists use various statistical methods to quantify and account for uncertainty in their results. This can include calculating confidence intervals, conducting multiple trials, and using error bars on graphs to visually represent uncertainty.

## 3. What are the sources of uncertainty in scientific measurements?

Sources of uncertainty can include limitations of measuring instruments, human error, variability in the phenomenon being studied, and external factors such as environmental conditions or interference from other variables.

## 4. How does uncertainty affect the reliability of scientific data?

Uncertainty can affect the reliability of scientific data by introducing potential bias or inaccuracies. It is important for scientists to carefully consider and address uncertainty in their methods and analysis to ensure the validity and reliability of their data.

## 5. Can uncertainty ever be completely eliminated in scientific measurements?

No, uncertainty can never be completely eliminated in scientific measurements. However, scientists can minimize uncertainty by using precise and accurate measurement techniques, conducting multiple trials, and accounting for potential sources of error.

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