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In mathematics, a cubic equation is an equation that involves a polynomial of degree three. This means that the highest power of the variable in the equation is three, typically in the form of \( ax^3 + bx^2 + cx + d = 0 \). In the exercise, the cubic equation was \( 2r^3 + r^2 - 63 = 0 \). Solving cubic equations often requires methods like factoring, synthetic division, or numerical methods. When simple trials show a solution, we can verify it by substituting back. Numerical methods, such as the use of a calculator, can also find roots efficiently if the solutions are not easy to spot. Cubic equations can have up to three real roots or a combination of real and complex roots, making them a bit tricker than quadratic equations.

A right circular cylinder is a 3D geometric shape characterized by two parallel circular bases and a straight line, or axis, connecting their centers at a right angle. It is like a soup can, perfectly round, and maintains the same diameter from top to bottom. The problem involves such a cylinder to calculate its volume, given a specific height-to-radius relationship. Understanding this shape involves visualizing its components:

• The height \( h \) is the distance between the bases.
• The radius \( r \) is the distance from the center to the edge of the circular base.

The relationship given in the problem, \( h = 1 + 2r \), shows how the height is expressed in terms of the radius. These dimensions are critical for calculating properties like area and volume.

The volume of a right circular cylinder can be determined using the formula \( V = \pi r^2 h \). This formula illustrates how the volume depends on both the radius and the height of the cylinder. The expression \( \pi r^2 \) represents the area of the base circle, and multiplying by the height \( h \) gives the total volume. Using the known volume \( 63\pi \) from the problem:

• First, substitute the volume into the formula: \( 63\pi = \pi r^2 h \).
• Then, simplify by dividing both sides by \( \pi \), leading to \( 63 = r^2 h \).

This equation shows the direct relationship between height and radius needed to solve for both dimensions. Understanding how each component of the formula relates to physical dimensions helps in solving real-world problems involving cylindrical objects, like calculating the capacity of containers or volumes of spaces.