To find the diameter of the cylinder, we will use 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,
- \( \pi \) is approximately 3.14.
We are given:
- \( V = 29,673 \) inches³,
- \( h = 42 \) cm.
First, we need to convert the height from centimeters to inches, since the volume is given in inches³. We know that: 1 inch = 2.54 cm, so:
\[ h = 42 \text{ cm} \times \frac{1 \text{ inch}}{2.54 \text{ cm}} \approx 16.54 \text{ inches} \]
Now we can substitute the volume and height into the volume formula and solve for the radius \( r \):
\[ 29,673 = 3.14 r^2 (16.54) \]
Calculating \( 3.14 \times 16.54 \):
\[ 3.14 \times 16.54 \approx 51.87 \]
Now our equation looks like this:
\[ 29,673 = 51.87 r^2 \]
Now, divide both sides by 51.87 to isolate \( r^2 \):
\[ r^2 = \frac{29,673}{51.87} \approx 572.57 \]
Now take the square root to find \( r \):
\[ r \approx \sqrt{572.57} \approx 23.94 \text{ inches} \]
The diameter \( d \) is twice the radius:
\[ d = 2r \approx 2 \times 23.94 \approx 47.88 \text{ inches} \]
Given the possible options and the rounded result for diameter, it appears I may have misunderstood your input. Out of the choices provided:
- 30 inches
- 15 inches
- 10 inches
- 5 inches
Based on our calculation:
\[ d \approx 48 \text{ inches} \]
None of the provided options match the outcome. Please ensure the inputs are correct, or check if other choices should be included.