Chapter 12.5, Problem 10WE

To determine

To find:The capacity of holding coal by the actual hopper car in cubic feet.

Answer to Problem 10WE

The capacity of holding coal by the actual hopper car in cubic feet is $5760{\text{ft}}^{\text{3}}$ .

Explanation of Solution

Given information:

The scale factor for model freight train is $1:48$ and amount of coal hold by model hopper caris $90\text{in}{\text{.}}^3$ .

Calculation:

The ratio of the corresponding sides of any two similar geometric figures is termed as Scale factor.

Let the total volume of the model hopper car be $V_1$ and the total volume of actual hopper car be $V_2$ and scale factor here for both the model be $s_1$ and $s_2$ .

The theorem of similar solids states that if the scale factor of two similar solids is $a:b$ , then the ratio of their volumes is $a^3:b^3$ .

Expression to determine the volume of actual hopper car from the above theorem is:

  $\begin{array}{c}\text{Volume}={\left(\text{scale}\text{factor}\right)}^3\\ \frac{V_1}{V_2}={\left(\frac{s_1}{s_2}\right)}^3\end{array}$

Substitute $1$ for $s_1$ , $48$ for $s_2$ and $90\text{in}{\text{.}}^{\text{3}}$ for $V_1$ in the above expression.

  $\begin{array}{c}\frac{90\text{in}{\text{.}}^{\text{3}}}{V_1}={\left(\frac{1}{48}\right)}^3\\ V_1=90\text{in}{\text{.}}^{\text{3}}\times 48\times 48\times 48\\ =9,953,280\text{in}{\text{.}}^{\text{3}}\end{array}$

Formula to convert cubic inches to cubic feet is:

  $1\text{in}{\text{.}}^{\text{3}}=\frac{1}{1728}{\text{ft}}^{\text{3}}$

To find $9,953,280\text{in}{\text{.}}^{\text{3}}$ in cubic feet, multiply both sides of the above expression by $9,953,280$ .

  $\begin{array}{c}9,953,280\times 1\text{in}{\text{.}}^{\text{3}}=9,953,280\times \frac{1}{1728}{\text{ft}}^{\text{3}}\\ =5760{\text{ft}}^{\text{3}}\end{array}$

Therefore, the capacity of holding coal by the actual hopper car in cubic feet is $5760{\text{ft}}^{\text{3}}$ .