How to Calculate the Empty Space in a Fountain: Step-by-Step Guide
Quick Answer
To find the empty space in a fountain, subtract the volume of the water from the total volume of the cylinder. First, calculate the cylinder's volume, then the hemisphere's volume, and finally, perform the subtraction.
Understanding how to calculate the empty space in a fountain involves a few straightforward steps. Let’s break it down together.
1. **Identify the Shapes**: In this case, your fountain consists of a solid concrete cylinder with a hemisphere scooped out to hold water. This means that to find the empty space, you need to know the volume of both the cylinder and the hemisphere.
2. **Calculate the Volume of the Cylinder**: To find the total volume of the cylinder, use the formula:
**V_cylinder = π r² h**
Here, the radius (r) is 8 inches (the part inside the fountain), and the height (h) is 12 inches. Plugging in these values:
V_cylinder = π (8 in)² (12 in)
V_cylinder = π (64 in²) (12 in)
V_cylinder = 768π in³
Approximating π as 3.14, the volume of the cylinder is about 2413.76 in³.
3. **Calculate the Volume of the Hemisphere**: For the hemisphere, use the formula:
**V_hemisphere = (2/3) π r³**
Using the same radius of 8 inches:
V_hemisphere = (2/3) π (8 in)³
V_hemisphere = (2/3) π (512 in³)
V_hemisphere = (1024/3) π in³
Approximating π again, the volume of the hemisphere is about 1076.48 in³, which matches your calculation of 1072.33 in³, assuming slight rounding differences.
4. **Find the Empty Space**: Now that you have both volumes, you can find the empty space (the concrete part of the fountain) by subtracting the volume of the water from the volume of the cylinder:
Empty Space = V_cylinder - V_hemisphere
Empty Space = 2413.76 in³ - 1076.48 in³
Empty Space ≈ 1337.28 in³.
5. **Real-World Application**: Understanding how to calculate the empty space in objects like fountains can be useful in various fields, including landscaping, construction, and manufacturing. Knowing the volume can help in estimating materials needed or understanding how much water your fountain can hold.
In summary, by calculating the volume of the entire cylinder and then subtracting the volume of the water (hemisphere), you can easily determine the empty space in your fountain. This method can be applied to various other geometric objects as well, making it a useful skill in both academic and practical settings.
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