Q34P - Fundamentals Of Physics

Question

Two types of barrel units were in use in the 1920s in the United States. The apple barrel had a legally set volume of 7056 cubic inches; the cranberry barrel, 5826 cubic inches. If a merchant sells 20 cranberry barrels of goods to a customer who thinks he is receiving apple barrels, what is the discrepancy in the shipment volume in liters?

Step-by-Step Solution

Answer

Less amount of volume received by the customer is${}^{\mathbf{403}\mathbf{ }\mathbf{liters}}$. 

Step 1: Given data

$\mathbf{1}\mathbf{ }\mathbf{apple}\mathbf{ }\mathbf{barrel}\mathbf{=}\mathbf{7056}\mathbf{ }{\mathbf{inch}}^{\mathbf{3}}$

$\mathbf{1}\mathbf{ }\mathbf{cranberry}\mathbf{ }\mathbf{barrel}\mathbf{=}{\mathbf{5826}}^{\mathbf{3}}$

Step 2: Understanding the conversion factor

Unit conversion is the conversion between different units of measure for the same quantity, usually through multiplier conversion factors. Here, apple barrel and cranberry are the different units for the quantity volume.

Step 3: Determination of the total volume in the cranberry barrel

Assume that the total volume in the cranberry barrel is ${}^{{\mathbf{V}}_{\mathbf{c}}}$ and the total volume in the apple barrel is ${}^{{\mathbf{V}}_{\mathbf{A}}}$ .

The total volume in the cranberry barrel is calculated as: 

$$\begin{gathered} {\mathbf{V}}_{\mathbf{c}}\mathbf{=}\mathbf{20}\mathbf{\times }\mathbf{ }\mathbf{Volume}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{cranberry}\mathbf{ }\mathbf{barrel} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{20}\mathbf{\times }\mathbf{5826}\mathbf{ }{\mathbf{inch}}^{\mathbf{3}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{11652}\mathbf{ }{\mathbf{inch}}^{\mathbf{3}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{116}\mathbf{.}\mathbf{52}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{ }{\mathbf{inch}}^{\mathbf{3}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ } \end{gathered}$$

Therefore, the total volume of the cranberry barrel is ${}^{\mathbf{116}\mathbf{.}\mathbf{52}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{ }{\mathbf{inch}}^{\mathbf{3}}}$

Step 4: Determination of the total volume in the apple barrel

The total volume in the apple barrel is calculated as: 

$$\begin{gathered} {\mathbf{V}}_{\mathbf{A}}\mathbf{=}\mathbf{20}\mathbf{\times }\mathbf{\times }\mathbf{Volume}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{apple}\mathbf{ }\mathbf{barrel} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{20}\mathbf{\times }\mathbf{7056}\mathbf{ }{\mathbf{inch}}^{\mathbf{3}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{141120}\mathbf{ }{\mathbf{inch}}^{\mathbf{3}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{141}\mathbf{.}\mathbf{12}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{ }{\mathbf{inch}}^{\mathbf{3}} \end{gathered}$$

Therefore, the total volume of the apple barrel is${}^{\mathbf{141}\mathbf{.}\mathbf{12}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{ }{\mathbf{inch}}^{\mathbf{3}}}$ .

Step 5: Determination of the discrepancy in the shipment volume

The discrepancy in the shipment volume received by the customer ${}^{(∆V)}$ is equal to the difference in the volumes of cranberry barrel and apple barrel. Therefore,

$$\begin{gathered} \mathbf{\Delta V}\mathbf{=}{\mathbf{V}}_{\mathbf{A}}\mathbf{-}{\mathbf{V}}_{\mathbf{C}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{141}\mathbf{.}\mathbf{12}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}\mathbf{ }}{\mathbf{inch}}^{\mathbf{3}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{24}\mathbf{.}\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{ }{\mathbf{inch}}^{\mathbf{3}} \end{gathered}$$

The discrepancy in volume in liters is, 

$$\begin{gathered} \mathbf{\Delta V}\mathbf{=}\mathbf{24}\mathbf{.}\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{ }{\mathbf{inch}}^{\mathbf{3}\mathbf{\times }}\mathbf{ }\frac{\mathbf{1}\mathbf{ }\mathbf{liter}}{\mathbf{61}\mathbf{.}\mathbf{0237}\mathbf{ }{\mathbf{inch}}^{\mathbf{3}}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{403}\mathbf{.}\mathbf{122}\mathbf{ }\mathbf{liters} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\approx }\mathbf{403}\mathbf{ }\mathbf{liters} \end{gathered}$$

Thus, the discrepancy in the shipment volume is ${}^{\mathbf{403}\mathbf{ }\mathbf{liters}}$ .