Calculating the Volume of a Cylinder with Hemispherical End Caps

In this article, we will explore the method to calculate the total volume of a cylinder with two hemispherical end caps. The given dimensions are a cylinder with a body length of 3 units and two hemispherical end caps each with a radius of 1 unit. We will break down the problem into simpler steps and use the appropriate formulas to arrive at the final answer.

Step 1: Volume of the Cylindrical Body

The formula for the volume of a cylinder is given by:

V_{text{cylinder}}  pi r^2 h

where r is the radius of the cylinder and h is the height or length of the cylinder. Given that the radius of the cylinder is 1 unit and the height is also 3 units, we can substitute these values into the formula:

V_{text{cylinder}}  pi 1^2 3  3pi

Step 2: Volume of the Hemispherical Caps

The volume of a sphere is given by:

V  frac{4}{3} pi r^3

Since each end cap is a hemisphere, the volume of one hemisphere is half of the volume of a sphere:

V_{text{hemisphere}}  frac{1}{2} cdot frac{4}{3} pi r^3  frac{2}{3} pi r^3

For a hemisphere with a radius of 1 unit:

V_{text{hemisphere}}  frac{2}{3} pi 1^3  frac{2}{3} pi

Since there are two hemispheres, the total volume of the hemispherical caps is:

V_{text{two hemispheres}}  2 cdot frac{2}{3} pi  frac{4}{3} pi

Step 3: Total Volume

Now, we can calculate the total volume by adding the volume of the cylindrical body and the volume of the two hemispherical caps:

V_{text{total}}  V_{text{cylinder}}   V_{text{two hemispheres}}  3pi   frac{4}{3}pi

To add these two values, we need a common denominator:

3pi  frac{9}{3} pi

Adding the two volumes:

V_{text{total}}  frac{9}{3} pi   frac{4}{3} pi  frac{13}{3} pi

Final Answer

The total volume of the cylinder with two hemispherical end caps is:

boxed{frac{13}{3} pi} text{ cubic units.}

Alternative Calculation

Another approach to calculating the volume is to consider the volume of the two hemispheres together making a whole sphere:

Volume of two end caps is 4/3 pi r^3  4/3 pi units of volume

Volume of the cylinder is pi r^2 h 3pi units of volume.

Adding both volumes, we get:

Total volume: 13pi/3 units of volume or about 13.6 units of volume.

Conclusion

By breaking the problem down into these steps, we can accurately determine the total volume of a cylinder with hemispherical end caps. The calculations provided in this article follow standard mathematical procedures and confirm the final result of 13pi/3 cubic units.

For further exploration, you might want to investigate similar geometrical problems or delve into more advanced volume calculations involving complex shapes.