A paraboloid is a three-dimensional shape that looks like a parabola rotated around its axis. It's often seen in satellite dishes and certain types of architectural structures. Calculating its volume involves understanding its geometric properties and applying a specific formula.
Volume Formula for a Paraboloid
The volume (\( V \)) of a paraboloid is given by the formula:
\[ V = \dfrac{1}{2} \cdot \pi \cdot a^2 \cdot h \]
Where:
- \( V \) is the volume of the paraboloid.
- \( a \) is the radius of the base.
- \( h \) is the height of the paraboloid.
- \( \pi \) is approximately equal to 3.14159.
This formula calculates the volume by integrating the area of circular cross-sections along the height of the paraboloid.
Step-by-Step Calculation
Let’s go through the calculation with an example.
Given:
- Radius of the base (\( a \)) = 5 units
- Height of the paraboloid (\( h \)) = 12 units
Step-by-Step Calculation
Step 1: Identify the Given Values
Given:
- \( a = 5 \) units
- \( h = 12 \) units
Step 2: Substitute Values into the Volume Formula
Using the formula:
\[ V = \dfrac{1}{2} \cdot \pi \cdot a^2 \cdot h \]
Substitute \( a = 5 \) and \( h = 12 \):
\[ V = \dfrac{1}{2} \cdot \pi \cdot 5^2 \cdot 12 \]
Step 3: Simplify the Expression Inside the Parentheses
Calculate each term:
\[ 5^2 = 25 \]
Step 4: Substitute and Simplify
Now substitute back into the volume formula:
\[ V = \dfrac{1}{2} \cdot \pi \cdot 25 \cdot 12 \]
Step 5: Calculate the Final Value
First, multiply \( 25 \cdot 12 \):
\[ 25 \cdot 12 = 300 \]
Then, multiply by \( \dfrac{1}{2} \):
\[ \dfrac{1}{2} \cdot 300 = 150 \]
Finally, multiply by \( \pi \):
\[ V = 150 \cdot \pi \]
Using \( \pi \approx 3.14159 \):
\[ V \approx 150 \cdot 3.14159 \]
\[ V \approx 471.239 \text{ cubic units} \]
Final Value
The volume of a paraboloid with a base radius of 5 units and a height of 12 units is approximately \( 471.239 \) cubic units.
