To find the diameter of the right circular cylinder, we can use the formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
We are given:
- Volume \( V = 29,673 \) cubic inches
- Height \( h = 42 \) cm
Before substituting the values into the formula, we need to convert the height from centimeters to inches. Since there are 2.54 centimeters in an inch, we convert the height as follows:
\[ h = \frac{42 \text{ cm}}{2.54 \text{ cm/in}} \approx 16.5354 \text{ inches} \]
Now we can plug the values into the volume formula:
\[ 29,673 = 3.14 r^2 (16.5354) \]
Now, we simplify and solve for \( r^2 \):
\[ 29,673 = 3.14 \cdot 16.5354 \cdot r^2 \] \[ 29,673 \approx 51.8416 r^2 \]
Next, we solve for \( r^2 \):
\[ r^2 = \frac{29,673}{51.8416} \approx 572.12 \]
Now, take the square root to find \( r \):
\[ r \approx \sqrt{572.12} \approx 23.94 \text{ inches} \]
Now to find the diameter \( d \):
\[ d = 2r \approx 2 \cdot 23.94 \approx 47.88 \text{ inches} \]
However, since the options provided (15 inches, 10 inches, 5 inches, and 30 inches) do not align with our result, we need to check our approach once more or review the assumed values, especially in unit conversions or calculations.
Since the calculated diameter does not conform to any offered responses, it's advised to verify the values provided or reconsider any assumptions about the values to match one of the options closely. Nonetheless, based on our calculated volume and height, the process yields a substantially larger diameter befits the calculated radius, indicating a misalignment with choices present. Thus ordinarily the choices presented in literal capacity wouldn't apply accurately here unless characteristics or scale corrections imply otherwise in the input ascertainment from the problem representation.