A barrel is a cylindrical container with bulging sides, typically made of wooden staves bound by metal hoops. In geometry, we consider it as a solid formed by rotating a curved line around an axis, creating a shape similar to two truncated cones joined at their wider ends.
Calculating the volume of a barrel involves using the formula for a truncated cone and accounting for the barrel's unique shape. We use the radii at the top, middle, and bottom of the barrel, along with its height, to determine the volume.
The formula for calculating the volume of a barrel is:
\[ V = \frac{\pi h}{12} (r_1^2 + 4r_2^2 + r_3^2) \]
Where:
Let's calculate the volume of a barrel with the following measurements:
Applying the formula:
\[ \begin{align*} V &= \frac{\pi h}{12} (r_1^2 + 4r_2^2 + r_3^2) \\ &= \frac{\pi \cdot 100}{12} (30^2 + 4 \cdot 40^2 + 30^2) \\ &= \frac{100\pi}{12} (900 + 6400 + 900) \\ &= \frac{100\pi}{12} (8200) \\ &\approx 214,675.5 \text{ cm}^3 \\ &\approx 214.68 \text{ liters} \end{align*} \]
Here's a visual representation of this barrel:
In this diagram, you can see the barrel shape with its key measurements labeled. The red line represents the top radius (r₁), the green line the middle radius (r₂), and the blue line the bottom radius (r₃). The purple line shows the height (h) of the barrel. The calculated volume is displayed below the barrel.
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