To find the uncertainty values for the area of each object, we'll use error propagation rules. The area $A$ of a circle is related to the diameter $D$ by the formula:

$A = \pi \left( \frac{D}{2} \right)^2 = \frac{\pi D^2}{4}$

The uncertainty in the area, $\Delta A$, can be determined by propagating the uncertainty in the diameter. For a function of the form $A = kD^2$ (where $k = \frac{\pi}{4}$), the uncertainty in $A$ is given by:

$\Delta A = 2kD \cdot \Delta D = \frac{\pi D}{2} \cdot \Delta D$

Step-by-Step Calculation:

For each object, we'll calculate the uncertainty in the area using the diameter and its uncertainty.

1. Object 1: Diameter $D = 9.7 \pm 0.1$ cm

$\Delta A_1 = \frac{\pi \cdot 9.7}{2} \cdot 0.1 = 1.524 \, \text{cm}^2$ So, the area uncertainty for object 1 is $\pm 1.52 \, \text{cm}^2$.

2. Object 2: Diameter $D = 6.7 \pm 0.1$ cm

$\Delta A_2 = \frac{\pi \cdot 6.7}{2} \cdot 0.1 = 1.052 \, \text{cm}^2$ So, the area uncertainty for object 2 is $\pm 1.05 \, \text{cm}^2$.

3. Object 3: Diameter $D = 6.9 \pm 0.1$ cm

$\Delta A_3 = \frac{\pi \cdot 6.9}{2} \cdot 0.1 = 1.083 \, \text{cm}^2$ So, the area uncertainty for object 3 is $\pm 1.08 \, \text{cm}^2$.

4. Object 4: Diameter $D = 5.0 \pm 0.1$ cm

$\Delta A_4 = \frac{\pi \cdot 5.0}{2} \cdot 0.1 = 0.785 \, \text{cm}^2$ So, the area uncertainty for object 4 is $\pm 0.79 \, \text{cm}^2$.

5. Object 5: Diameter $D = 16.0 \pm 0.1$ cm

$\Delta A_5 = \frac{\pi \cdot 16.0}{2} \cdot 0.1 = 2.513 \, \text{cm}^2$ So, the area uncertainty for object 5 is $\pm 2.51 \, \text{cm}^2$.

Final Table with Uncertainties in Area:

Object	Diameter (cm)	Circumference (cm)	Area (cm²)
1	$9.7 \pm 0.1$	$30.47 \pm 0.314$	$73.90 \pm 1.52$
2	$6.7 \pm 0.1$	$21.05 \pm 0.314$	$35.26 \pm 1.05$
3	$6.9 \pm 0.1$	$21.68 \pm 0.314$	$37.39 \pm 1.08$
4	$5.0 \pm 0.1$	$15.71 \pm 0.314$	$19.63 \pm 0.79$
5	$16.0 \pm 0.1$	$50.27 \pm 0.314$	$201.06 \pm 2.51$

Would you like more details or have any questions?

Here are 5 related questions for deeper understanding:

1. How does the uncertainty in circumference relate to that in diameter?
2. Why is the uncertainty in area larger than that in diameter?
3. What role does the constant $\pi$ play in propagating uncertainties?
4. How would a larger diameter uncertainty affect the final results?
5. Can the uncertainties in the measurements be minimized, and how?

Tip: When propagating uncertainties, always consider how each measured variable affects the final calculated result to avoid underestimating the total uncertainty.