91Ó°ÊÓ

In geometry, inequalities are crucial for understanding the relationships between different geometric objects. When we deal with inequalities involving squares, like this one \(x^2 + y^2 - 14y + z^2 \geq -13\), it means we are not only considering the exact position, but a range of positions around it.
Inequalities help us identify regions rather than single points or lines. For instance, the inequality could define a region consisting of all points that satisfy the conditions given by the inequality.

• "Greater than or equal to" implies inclusion of the boundary.
• "Less than" or "greater than" alone would imply excluding the boundary.

Thus, geometric inequalities guide us to understand shapes like spheres being within or outside a certain region. They are fundamental in solving real-world problems, especially in fields involving spatial analysis like physics and computer graphics.

A sphere in three-dimensional space is a perfect example of a geometrical shape that we often work with. The general equation for a sphere is \((x-h)^2+(y-k)^2+(z-l)^2 = r^2\), where

• \((h, k, l)\) is the center of the sphere.
• \(r\) is the radius of the sphere.
• "\(=\)" means only the surface of the sphere.

When inequalities are involved, like \(\geq\), they tell us that we should include not just the surface, but everything inside the sphere too.

In our exercise, \(x^2+(y-7)^2+z^2 \geq 36\) transforms this into a situation where we consider every point in and on the sphere, including its center point \((0, 7, 0)\) and extending outward up to a distance (radius) of 6.

Completing the square is a mathematical technique used to solve, simplify, or rewrite quadratic equations. This process involves rearranging a quadratic equation into a form that is easier to work with or visually understand.

In our context, we started with \(x^2 + y^2 - 14y + z^2 \), noticed the \(y-14y\) term, and decided to complete the square for it. This means adjusting the equation to make \((y-7)^2\) appear, which represents a perfectly squared binomial.

To do this, we added and subtracted 49, leading to \(x^2 + (y-7)^2 + z^2 \geq 36\).

• This makes the equation easier to interpret as it forms a perfect square trinomial.
• This process can simplify graphing, solving, or understanding the geometry involved.

Completing the square not only provides clarity but also helps to transition problems from abstract algebraic expressions to clear geometric interpretations, like illustrating the dimensions of a sphere in space.