To find the diameter of the cylinder, we first need the formula for the volume of a right circular cylinder:
\[ V = \pi r^2 h \]
where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height, and
- \( \pi \) is a constant (which we'll use as 3.14).
Given:
- Volume \( V = 29,673 , \text{inches}^3 \)
- Height \( h = 42 , \text{cm} \)
First, we need to convert the height from centimeters to inches. Since 1 inch = 2.54 cm:
\[ h , \text{in inches} = \frac{42 , \text{cm}}{2.54 , \text{cm/inch}} \approx 16.54 , \text{inches} \]
Now, we can substitute the values into the volume formula:
\[ 29,673 = 3.14 r^2 \cdot 16.54 \]
Next, solve for \( r^2 \):
\[ 29,673 = 52.0386 r^2 \]
\[ r^2 = \frac{29,673}{52.0386} \approx 570.35 \]
Taking the square root to find \( r \):
\[ r \approx \sqrt{570.35} \approx 23.88 , \text{inches} \]
To find the diameter (which is twice the radius):
\[ \text{Diameter} = 2r \approx 2 \cdot 23.88 \approx 47.76 , \text{inches} \]
Since none of the provided responses were close to the calculated diameter, it seems there might be confusion with either the numbers or the units used. However, please ensure conversions are done accurately as unit discrepancies (inches vs. centimeters) significantly affect calculations.
Based on the provided information, the diameter does not match the available options. If these are indeed the options provided, it would be necessary to check if there's any error in the problem statement or expected outcomes.