To solve this problem, you want to calculate the correct average diameter of the solid sphere and its uncertainty based on the given measurements. The formula for calculating the uncertainty (standard deviation of the mean) is typically derived from statistical principles. Let's break this down step by step.

Step 1: Calculate the Mean Diameter

Given measurements of the sphere's diameter:

• 8.15 cm
• 8.10 cm
• 8.00 cm
• 8.15 cm
• 8.00 cm

The mean $\bar{x}$ is given by:

$\bar{x} = \frac{\sum x_i}{n}$

Where $x_i$ are the measurements, and $n = 5$ (the number of measurements).

$\bar{x} = \frac{8.15 + 8.10 + 8.00 + 8.15 + 8.00}{5} = \frac{40.40}{5} = 8.08 \, \text{cm}$

Step 2: Calculate the Standard Deviation

The formula for the standard deviation $s$ is:

$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

First, compute $(x_i - \bar{x})^2$ for each measurement:

• For $x_1 = 8.15$, $(8.15 - 8.08)^2 = 0.0049$
• For $x_2 = 8.10$, $(8.10 - 8.08)^2 = 0.0004$
• For $x_3 = 8.00$, $(8.00 - 8.08)^2 = 0.0064$
• For $x_4 = 8.15$, $(8.15 - 8.08)^2 = 0.0049$
• For $x_5 = 8.00$, $(8.00 - 8.08)^2 = 0.0064$

Now, sum the squared deviations:

$\sum (x_i - \bar{x})^2 = 0.0049 + 0.0004 + 0.0064 + 0.0049 + 0.0064 = 0.023$

Then calculate the standard deviation:

$s = \sqrt{\frac{0.023}{5-1}} = \sqrt{\frac{0.023}{4}} = \sqrt{0.00575} = 0.0758$

Step 3: Calculate the Uncertainty (Standard Error of the Mean)

The uncertainty $\Delta x$ (standard error of the mean) is given by:

$\Delta x = \frac{s}{\sqrt{n}} = \frac{0.0758}{\sqrt{5}} = \frac{0.0758}{2.236} \approx 0.034 \, \text{cm}$

Step 4: Final Answer

The correct way to write the diameter with its uncertainty is:

$(8.08 \pm 0.034) \, \text{cm}$

This corresponds to option e.

Do you have any questions or need further details? Here are some related questions:

1. How would the uncertainty change if more measurements were taken?
2. Why do we use $n-1$ in the standard deviation formula?
3. How can you interpret the significance of the uncertainty value?
4. What are other ways to measure uncertainty in experimental data?
5. What if one of the measurements was significantly different, how would that impact the results?

Tip: Always double-check your calculations for the mean and standard deviation, as small errors can impact the final uncertainty result.