Q51P - Fundamentals Of Physics

Question

The cubit is an ancient unit of length based on the distance between the elbow and the tip of the middle finger of the measurer. Assume that the distance ranged from ${}^{43 \mathrm{to} 53 \mathrm{cm}}$ , and suppose that ancient drawings indicate that a cylindrical pillar was to have a length of ${}^{9 \mathrm{cubits}}$ and a diameter of . For the stated range, what are the lower value and the upper value, respectively${}^{2 \mathrm{cubits}}$ , for (a) the cylinder’s length in meters, (b) the cylinder’s length in millimeters, and (c) the cylinder’s volume in cubic meters?

Step-by-Step Solution

Answer

1. The cylinder’s length in meters for lower value is 3.9 m and for upper value is 4.8 m 
2. The cylinder’s length in millimeters for lower value is $3.9\times {10}^3$and for upper value is $4.8\times {10}^3 \mathrm{mm}$
3. The volume of cylinder in cubic meters for lower value is   $2.2 {\mathrm{meters}}^3$and for upper value is$4.2 meter{s}^3$ .

Step 1: Given data

The range of the distance for cubit is from 43 to 53 cm.

The length of the cylindrical pillar is 9 cubits.

The diameter of the pillar is 2 cubits. 

Step 2: Understanding the unit conversion

Chain-link conversions are used for unit conversion in which given values are multiplied successively by conversion factors.

Step 3: (a) Determination of the length of the cylinder in meters

For lower value, the length of 9 cubits is,

$$\begin{gathered} 9 \mathrm{cubit} \mathrm{s}= 9 \mathrm{cubits}.\left(\frac{43 \mathrm{cm}}{1 \mathrm{cubits}}\right)\left(\frac{{10}^{-2} \mathrm{m}}{1 \mathrm{cm}}\right) \\ =3.87 \mathrm{m} \\ \approx 3.9 \mathrm{m} \end{gathered}$$

Thus, cylinder’s length for lower value is 3.9 meters . 

 

For higher value, the length of 9 cubits is, 

 

 $$\begin{gathered} 9 \mathrm{cubit} \mathrm{s}= 9 \mathrm{cubits}.\left(\frac{53 \mathrm{cm}}{1 \mathrm{cubits}}\right)\left(\frac{{10}^{-2} \mathrm{m}}{1 \mathrm{cm}}\right) \\ =4.77 \mathrm{m} \\ \approx 4.8 \mathrm{m} \end{gathered}$$

 

Thus, cylinder’s length for higher value is 4.8 m .

Step 4: (b) Determination of the length of the cylinder in mm

Length of cylinder for lower values in mm is calculated as, 

$$\begin{gathered} 3.9 \mathrm{m}=3.9.\left(\frac{{10}^3 \mathrm{mm}}{1 \mathrm{m}}\right) \\ =3.9\times {10}^{3 }\mathrm{mm} \end{gathered}$$

Thus, the length of cylinder for lower value is $3.9\times {10}^3\mathrm{mm}$ .

Length of cylinder for higher values in mm is calculated as,

$$\begin{gathered} 4.8 \mathrm{m}=4.8.\left(\frac{{10}^3 \mathrm{mm}}{1 \mathrm{m}}\right) \\ =4.8\times {10}^3\mathrm{mm} \end{gathered}$$

Thus, the length of cylinder for higher value is $4.8\times {10}^3\mathrm{mm}$.

Step 5: (c) Determination of the volume of the cylinder

Find the volume for both lower value and higher value.

First, converts the diameter into meters for lower value.

$$\begin{gathered} 2 \mathrm{cubit}=2 \mathrm{cubits}.\left(\frac{43 \mathrm{cm}}{1 \mathrm{cubit}}\right)\left(\frac{{10}^{-2} \mathrm{m}}{1 \mathrm{cm}}\right) \\ =0.86 \mathrm{m} \end{gathered}$$

Converts the diameter into meters for higher value.

$$\begin{gathered} 2 \mathrm{cubit}=2 \mathrm{cubits}.\left(\frac{53 \mathrm{cm}}{1 \mathrm{cubit}}\right)\left(\frac{{10}^{-2} \mathrm{m}}{1 \mathrm{cm}}\right) \\ =1.06 \mathrm{m} \end{gathered}$$

The expression for the volume of cylinder is given as: