Problem 40 - College Physics

Question

In the Bible, Noah is instructed to build an ark 300 cubits long, 50.0 cubits wide, and 30.0 cubits high (vFig. 1.19). Historical records indicate a cubit is equal to half a yard. (a) What would be the dimensions of the ark in meters? (b) What would be the ark's volume in cubic meters? To approximate, assume that the ark is to be rectangular.

Step-by-Step Solution

Answer

The ark's dimensions are 137.16m x 22.86m x 13.716m, with a volume of 42873.56m³.

Step 1: Convert Dimensions from Cubits to Yards

A cubit is stated to be equivalent to half a yard. Therefore, we can convert the dimensions of the ark from cubits to yards. The ark's length is 300 cubits, width is 50 cubits, and height is 30 cubits. - Length in yards: \(300 \times 0.5 = 150\) yards - Width in yards: \(50 \times 0.5 = 25\) yards - Height in yards: \(30 \times 0.5 = 15\) yards

Step 2: Convert Dimensions from Yards to Meters

We know that 1 yard is equal to 0.9144 meters. Now we convert the dimensions of the ark from yards to meters. - Length in meters: \(150 \times 0.9144 = 137.16\) meters - Width in meters: \(25 \times 0.9144 = 22.86\) meters - Height in meters: \(15 \times 0.9144 = 13.716\) meters

Step 3: Calculate the Volume in Cubic Meters

The volume of a rectangular prism is calculated by multiplying its length, width, and height. Now, using the dimensions in meters:- Volume in cubic meters: \[137.16 \times 22.86 \times 13.716 = 42873.56\] cubic meters.

Key Concepts

Unit Conversion

Unit conversion is an essential skill when solving physics problems, especially those involving dimensions like length, area, and volume. In our example, Noah's Ark dimensions need conversion from cubits to a more universal unit, meters. Let's understand the conversion process:

First, historical data informs us that a cubit is equal to half a yard. Therefore, to convert the ark's measurements from cubits to yards, we multiply by 0.5:

• Length: 300 cubits × 0.5 = 150 yards
• Width: 50 cubits × 0.5 = 25 yards
• Height: 30 cubits × 0.5 = 15 yards

Now, with the ark's dimensions in yards, the next step is converting yards into meters. The conversion factor for yards to meters is that one yard equals 0.9144 meters. To find the dimensions in meters:

• Length: 150 yards × 0.9144 = 137.16 meters
• Width: 25 yards × 0.9144 = 22.86 meters
• Height: 15 yards × 0.9144 = 13.716 meters

This systematic approach using conversion factors makes the process straightforward and ensures consistent and accurate results.

Volume Calculation

Volume calculation is crucial when dealing with three-dimensional objects. A rectangular prism's volume is found by multiplying its length, width, and height. With the ark's dimensions converted to meters, we can easily calculate its volume:

To recall, the dimensions of Noah's ark in meters are:

• Length: 137.16 meters
• Width: 22.86 meters
• Height: 13.716 meters

We calculate the volume using the formula for the volume of a rectangular prism:
\[ Volume = ext{Length} \times ext{Width} \times ext{Height} \] Plug the dimensions into the formula:
\[ Volume = 137.16 \times 22.86 \times 13.716 \approx 42873.56 \text{ cubic meters} \]

This calculation shows that the ark has a very large capacity, which aligns with its legendary task of preserving numerous animal species.

Dimensional Analysis

Dimensional analysis involves checking the relationships between different physical quantities by identifying their base dimensions, which ensures consistency and accuracy in calculations. It's particularly useful for verifying units in physics problems.

When converting and calculating dimensions and volume, such as with Noah's Ark, employing dimensional analysis can guide us to ensure calculations are performed correctly. By understanding the dimensions involved:

• Length (L)
• Width (W)
• Height (H)

We check the consistency in units while calculating volume. Since volume requires units of length cubed (e.g., meters cubed), ensure that all dimension units are appropriately converted to meters before performing calculations. Hence, the arithmetic should remain dimensionally appropriate throughout.

Errors can occur if even one unit is off, but by following the principles of dimensional analysis, one can reliably verify and confirm that the solution remains consistent across all steps.