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A cylinder is a 3D shape featuring two identical flat ends that are circular, joined by a curved surface. The main dimensions of a cylinder include its radius, depicted by the symbol \( x \), and its height, denoted by \( y \). These parameters play a crucial role in defining the cylinder's geometry. The radius \( x \) represents the distance from the center to the edge of the circular base, while the height \( y \) indicates the distance between the two bases.
To find the volume of a cylinder, we use the formula \( V_{cylinder} = \pi x^2 y \). This involves calculating the area of the base (\( \pi x^2 \)) and multiplying it by the height \( y \). This formula is derived from the fundamental principle of stacking circular disks from the bottom to the top of the cylinder.

• The term \( \pi x^2 \) refers to the area of the base circle.
• Multiplying by the height \( y \) provides the volume, essentially counting the total number of these circle areas stacked to the cylinder's height.

A hemisphere represents half of a sphere and forms a 3D shape bounded by a circular plane and a curved surface. Hemispheres are applied in various real-world scenarios, such as in designing tanks or domes.
The radius of a hemisphere, symbolized as \( x \), is crucial in calculating its volume. The formula employed is \( V_{hemisphere} = \frac{2}{3}\pi x^3 \). This equation results from halving the formula for the volume of a full sphere, which is \( \frac{4}{3}\pi x^3 \).

• Simply put, the hemisphere is half of a sphere, hence the \( \frac{2}{3} \) factor.
• Multiplying \( \frac{4}{3}\pi x^3 \) by \( \frac{1}{2} \) simplifies to \( \frac{2}{3}\pi x^3 \), giving us the volume for one hemisphere.

In scenarios involving two hemispheres, like the oxygen tank example, the combined volume is doubled: \( 2 \times \frac{2}{3}\pi x^3 = \frac{4}{3}\pi x^3 \). Consequently, two hemispheres equate to the volume of a full sphere.

Volume formulas help calculate the space inside a 3D object, crucial for understanding capacity and the object's scale. In our exercise, the oxygen tank combines the volume of a cylinder and two hemispheres. These sections are unified to define the tank's total volume.
The composite formula for the total volume, given both elements, is \[ V(x, y) = \pi x^2 y + \frac{4}{3}\pi x^3 \]
This unified expression comprises two elements:

• \( \pi x^2 y \): representing the volume of the cylindrical section.
• \( \frac{4}{3}\pi x^3 \): revealing the volume contributed by the two hemispheres, equivalent to a full sphere.

Solving \( V(10,2) \) by substituting \( x = 10 \) and \( y = 2 \) yields an exact volume calculation corresponding to the specific parameter values. Each term reflects a distinct section of the tank, aiding in a detailed understanding of how changes in \( x \) or \( y \) impact the tank's volume overall.