Uncertainity and Error Question

[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!

 

[inno87]

Anyone have an answer to this? I'm still very unstable about this uncertainty question.

 

[Moetasim]

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.