91Ó°ÊÓ

The given data can be listed below,

• The uncertainty in the diameter is, ${\mathrm{D}}_0=Â\pm 0.02\mathrm{cm}.$
• The uncertainty in the thickness is, ${\mathrm{h}}_0=Â\pm 0.005\mathrm{cm}.$
• The maximum radius of the circular disk is, $\mathrm{r}=\frac{8.52\mathrm{cm}}{2}=4.26\mathrm{cm}.$
• The maximum thickness of the cookie is, ${\mathrm{h}}_{\mathrm{m}}=0.055\mathrm{cm}.$

Uncertainty measurements are quantized values of the error that appears in the mathematical parameters due to some unidentifiable flaw in the tools or limit of precision of tools used for the measurements.

Uncertainty can be of two types: Absolute and Relative. No instrument can be ideal and so it is important to know the percentage error to determine.

The formula for the average volume of a circular disk is given by,

$V_A=\mathrm{Ï€}{r_A}^2{h}_A$

Here, r is the radius of the disk, and his the thickness of the disk.

Substitute the values in the above equation,

Thus, the average volume of the circular cookie is $2.837{\text{cm}}^{\text{3}}$.

The value of thickness could not exceed the maximum value so the maximum volume of the disk can be given by,

$V_m=\mathrm{Ï€}{r_m}^2{h}_m$

Here, $h_m$is the maximum value of the thickness.

Substitute values in the above,

The volume could be small, so smaller volume can be given by,

$V_s=\mathrm{Ï€}{r_s}^2{h}_s$

Here, $r_s$is the radius of the disk and $h_s$is the thickness of the disk

Substitute values in the above,

The uncertainty in the volume is calculated as,

$V_0=V_AÂ\pm V_S$

Substitute the values in the above equation.

Thus, the uncertainty in the volume of cookie is $Â\pm 0.03{\text{cm}}^3$.

The ratio of the average diameter to the average thickness is given by,

$$\begin{gathered} \frac{{\mathrm{d}}_{\mathrm{A}}}{{\mathrm{h}}_{\mathrm{A}}}=\frac{8.50\mathrm{cm}}{0.050\mathrm{cm}} \\ =170:1 \end{gathered}$$

Thus, the ratio of the average diameter to the average thickness is$170:1.$

By taking the largest possible value of the diameter and the smallest possible thickness largest possible value for this ratio is given by

role="math" localid="1655791096826" $$\begin{gathered} \frac{{\mathrm{d}}_{\mathrm{L}}}{{\mathrm{h}}_{\mathrm{L}}}=\frac{8.52\mathrm{cm}}{0.045\mathrm{cm}} \\ =190:1 \end{gathered}$$

The smallest possible value of the ratio is given by,

$$\begin{gathered} \frac{{\mathrm{d}}_{\mathrm{s}}}{{\mathrm{h}}_{\mathrm{s}}}=\frac{8.48\mathrm{cm}}{0.055\mathrm{cm}} \\ =150:1 \end{gathered}$$

From the above expressions, it is concluded that uncertainty in the ratio of the diameter and thickness is $Â\pm 20.$

Thus, the ratio of the diameter and the thickness including the uncertainty is$\left(170Â\pm 20\right):1$