Let's go through this step-by-step and ensure that we adhere to the proper rules for significant figures:

Given Data:

• Height (h) = 2.71 cm
• Diameter (d) = 11.68 mm

First, let's convert everything to consistent SI units (meters).

1. Convert height to meters: $h = 2.71 \, \text{cm} = 2.71 \times 10^{-2} \, \text{m}$ This conversion maintains 3 significant figures, because the original measurement of height (2.71 cm) has 3 significant figures.

2. Convert diameter to meters: $d = 11.68 \, \text{mm} = 11.68 \times 10^{-3} \, \text{m}$ The diameter value has 4 significant figures.

3. Radius (r) of the cylinder: The radius is half of the diameter: $r = \frac{d}{2} = \frac{11.68 \times 10^{-3}}{2} = 5.84 \times 10^{-3} \, \text{m}$ The radius should have 3 significant figures because the diameter has 4 significant figures, but when divided by 2, it is limited by the least number of significant figures.

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Part a: Ratio of the Cylinder's Radius to its Height

The ratio of the radius $r$ to the height $h$ is: $\text{Ratio} = \frac{r}{h} = \frac{5.84 \times 10^{-3}}{2.71 \times 10^{-2}} = 0.215$ When performing this division, the result is limited by the number of significant figures in the measurements. The height has 3 significant figures, and the radius has 3 significant figures, so the ratio should have 3 significant figures.

Thus, the ratio is: $\boxed{0.215}$

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Part b: Volume of the Cylinder

The formula for the volume of a cylinder is: $V = \pi r^2 h$ Substitute the values: $V = \pi (5.84 \times 10^{-3})^2 \times 2.71 \times 10^{-2}$ Let's calculate it step by step:

1. Square the radius: $(5.84 \times 10^{-3})^2 = 3.41 \times 10^{-5} \, \text{m}^2$

2. Multiply by the height: $3.41 \times 10^{-5} \times 2.71 \times 10^{-2} = 9.23 \times 10^{-7} \, \text{m}^3$

3. Multiply by $\pi$: $V = \pi \times 9.23 \times 10^{-7} \approx 2.90 \times 10^{-6} \, \text{m}^3$

The volume value should be given with 3 significant figures (as the height and radius had 3 significant figures each).

Thus, the volume is: $\boxed{2.90 \times 10^{-6}} \, \text{m}^3$

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Summary of Results:

• a. Ratio of the radius to the height: $0.215$
• b. Volume of the cylinder: $2.90 \times 10^{-6} \, \text{m}^3$

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Do you need further details on these calculations or have additional questions?

Related Questions:

1. How would the volume change if the height were increased by 10%?
2. What would the radius be if the volume was doubled, keeping the height the same?
3. How does the number of significant figures affect the precision of the result?
4. How can we apply these measurements to calculate surface area?
5. How would the ratio of the radius to height change if the diameter were measured with a higher precision instrument?

Tip: Always ensure that the final result is limited by the measurement with the least number of significant figures when performing calculations.