A pressure vessel is shaped like a cylinder with hemispherical ends, operating at 1,100 kPa. If the overall length is 12 m and the diameter is 3 m, what is the volume of the vessel?

To determine the volume of a pressure vessel shaped like a cylinder with hemispherical ends, it's important to calculate the volumes of both the cylindrical section and the hemispherical ends, then combine these volumes to find the total.

The volume of the cylindrical section can be calculated using the formula for the volume of a cylinder, which is given by:

[ V_{cylinder} = \pi r^2 h ]

where ( r ) is the radius and ( h ) is the height (length) of the cylinder. In this case, the diameter of the vessel is 3 m, which gives a radius of ( 1.5 , \text{m} ) (since radius is half of the diameter). The length of the cylindrical section is ( 12 , \text{m} ).

Next, the volume of each hemispherical end is given by the formula for the volume of a sphere divided by 2, which is:

[ V_{hemisphere} = \frac{2}{3} \pi r^3 ]

Since there are two hemispherical ends, the total volume contributed by both hemispheres is twice this value.

Now, calculate the volumes:

1. **Volume of the Cylinder: