Standard errors in surface areas and volumes?

• zejaa7
In summary, the conversation revolves around a problem involving a copper cylinder with given measurements and finding the total surface area, volume, and mass of the cylinder. The individual is struggling with using error propagation formulas and is seeking guidance. The expert suggests using a specific formula and explains how it can be applied to the problem.
zejaa7
I have to finish this one question that I have come across and I am having a bit of trouble figuring out where to go from what I havee done.
The Q is:

A copper cylinder is 5.82 +/- 0.06 cm long and has a radius of 2.53 +/- 0.04 cm. Using the appropraite formula,
Question Details
a) Find the total surface area of the cylinder and the standard error in the area .
b) Find the volume of the cylinder and the standard error in the volume.
c) Given that the density of copper is 8.96cm^-3, find the mass of the cylinder

We are to use error propogation formulas for this.

For part a and b I calculated the normals and for b- volume and used the formulae for product rules to calculate the error in volume and found it to be 2.88.

Can someone guide me through the process because I have the formulas and I plug in the values but I am not sure If i am right or not.

Thank you.

CompuChip said:
Technically, the correct formula from error analysis is

$$\delta f(x_1, \cdots, x_n) = \sqrt{ \sum_{i = 1}^n \left( \frac{\partial f(x_1, \cdots, x_n)}{\partial x_i} \delta x_i \right)^2 }$$
where $\delta x_i$ is the uncertainty in xi.

I prefer to remember just this formula and see how it applies in any particular problem. In this case, for example, the volume of the cilinder would be given by $V = \pi r^2 \ell$ where r is the radius and $\ell$ its length.
Since both of these have an uncertainty associated to them, you can calculate
$$\frac{\partial V}{\partial \ell} = \pi r^2, \quad \frac{\partial V}{\partial r} = 2 \pi r \ell$$
and simply plug in your values for $r, \ell, \delta r \text{ and } \delta\ell$.

What are standard errors in surface areas and volumes?

Standard errors in surface areas and volumes are measures of the variability or uncertainty in the estimation of these quantities. They provide a range of values that is likely to contain the true surface area or volume with a certain level of confidence.

How are standard errors calculated for surface areas and volumes?

Standard errors for surface areas and volumes are calculated using mathematical formulas that take into account the sample size, the variability of the measurements, and the level of confidence desired. These calculations can be complex and may vary depending on the specific situation.

What is the significance of standard errors in surface areas and volumes?

Standard errors are important because they allow us to determine the precision of our measurements and the level of confidence we can have in our estimates. They also help us compare different measurements and determine if they are significantly different from each other.

How do standard errors differ from standard deviations?

Standard errors and standard deviations are both measures of variability, but they have different purposes. Standard deviations measure the variability of individual data points within a sample, while standard errors measure the variability of an estimated value. Standard errors are typically smaller than standard deviations because they take into account the sample size.

How can standard errors be used in real-world applications?

Standard errors can be used in various scientific and research applications, such as in the estimation of population parameters, in the evaluation of experimental results, and in the comparison of different measurements. They can also be used to determine the sample size needed for a study to achieve a certain level of precision and confidence.

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