To find the diameter of the 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.
First, we need to convert the height from centimeters to inches since the volume is given in cubic inches. There are 2.54 centimeters in an inch, so:
\[ h = 42 \text{ cm} \times \frac{1 \text{ in}}{2.54 \text{ cm}} \approx 16.54 \text{ in} \]
Now, we can rearrange the volume formula to solve for the radius \( r \):
\[ r^2 = \frac{V}{\pi h} \]
Substituting the volume and height into the equation:
\[ r^2 = \frac{29673}{3.14 \times 16.54} \]
Calculating the denominator:
\[ 3.14 \times 16.54 \approx 51.9276 \]
Now calculating \( r^2 \):
\[ r^2 = \frac{29673}{51.9276} \approx 571.64 \]
Taking the square root to find \( r \):
\[ r \approx \sqrt{571.64} \approx 23.9 \text{ inches} \]
Now, to find the diameter \( d \):
\[ d = 2r \approx 2 \times 23.9 \approx 47.8 \text{ inches} \]
Since none of the multiple-choice options (10 inches, 30 inches, 15 inches, 5 inches) matches the calculated diameter, it seems there might be a misinterpretation of the problem or wrong data provided. Please verify the height or volume details, or if calculations yield an unexpected result.