The problem asks to find the volume of a propane fuel tank that is cylindrical with hemispherical ends. The dimensions provided are:

• Total length of the tank (including hemispherical ends): 46 inches
• Diameter of the tank: 12 inches (radius = 12/2 = 6 inches)

The tank is made up of two parts:

1. A cylindrical part in the middle.
2. Two hemispherical ends.

Step 1: Volume of the cylindrical part

The volume of a cylinder is given by the formula: $V_{\text{cylinder}} = \pi r^2 h$

• $r = 6$ inches (radius of the cylinder)
• The length of the cylindrical part is the total length minus the length of the two hemispheres. Since each hemisphere has a radius of 6 inches, the total contribution of the hemispheres is $2 \times 6 = 12$ inches. Thus, the length of the cylindrical part is: $h = 46 - 12 = 34 \text{ inches}$ Now, calculate the volume of the cylindrical part: $V_{\text{cylinder}} = \pi (6)^2 (34) = \pi \times 36 \times 34 = 1224\pi \, \text{cubic inches}$ Using $\pi \approx 3.1416$: $V_{\text{cylinder}} \approx 3844.99 \, \text{cubic inches}$

Step 2: Volume of the two hemispherical ends

The volume of a sphere is given by the formula: $V_{\text{sphere}} = \frac{4}{3} \pi r^3$ Since we have two hemispheres (which together make a full sphere), we can calculate the volume of a full sphere: $V_{\text{sphere}} = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi \times 216 = \frac{864\pi}{3} = 288\pi \, \text{cubic inches}$ Using $\pi \approx 3.1416$: $V_{\text{sphere}} \approx 904.78 \, \text{cubic inches}$

Step 3: Total volume in cubic inches

The total volume of the tank is the sum of the volume of the cylinder and the sphere: $V_{\text{total}} = V_{\text{cylinder}} + V_{\text{sphere}} = 1224\pi + 288\pi = 1512\pi \, \text{cubic inches}$ Using $\pi \approx 3.1416$: $V_{\text{total}} \approx 4749.77 \, \text{cubic inches}$

Step 4: Convert to gallons

There are approximately 231 cubic inches in a gallon, so the volume in gallons is: $V_{\text{gallons}} = \frac{4749.77}{231} \approx 20.56 \, \text{gallons}$

Final Answers:

• Volume in cubic inches: $4749.77 \, \text{in}^3$
• Volume in gallons: $20.56 \, \text{gallons}$

Would you like more details or explanations on any step?

Here are 5 related questions to deepen your understanding:

1. How would the volume change if the radius was increased by 2 inches?
2. What if the tank was a full sphere instead of having cylindrical parts?
3. How is the volume of a cylinder different from the volume of a prism with the same base?
4. How would you convert the volume to liters instead of gallons?
5. How does changing the length of the cylindrical part affect the total volume?

Tip: When converting between units of volume (like cubic inches to gallons), always remember to keep track of unit conversions to avoid errors.