Problem 5 - Active Calculus Multivariable

Question

Find the equation of the sphere centered at (10,10,4) with radius \(9 .\) Normalize your equations so that the coefficient of \(x^{2}\) is \(1 .\) _________=0 Give an equation which describes the intersection of this sphere with the plane \(z=5\). _________=0

Step-by-Step Solution

Answer

The equation of the sphere is \(x^2 + y^2 + z^2 - 20x - 20y - 8z + 216 = 0\). The equation describing the intersection of the sphere with the plane \(z=5\) is \(x^2 + y^2 - 20x - 20y + 201 = 0\).

Step 1: Write down the general equation of the sphere

First, we will write down the general equation of the sphere without substituting our center and radius values: \[(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\]

Step 2: Substitute the center and radius values

Now we can substitute the values of center (10,10,4) and radius 9 into the general equation of the sphere: \[(x-10)^2 + (y-10)^2 + (z-4)^2 = 9^2\]

Step 3: Simplify the equation

Next, we need to simplify the equation and ensure that the coefficient of \(x^2\) is 1: \[x^2 - 20x + 100 + y^2 - 20y + 100 + z^2 - 8z + 16 = 81\] \[x^2 + y^2 + z^2 - 20x - 20y - 8z + 216 = 0\] So we have the equation of the sphere: \[x^2 + y^2 + z^2 - 20x - 20y - 8z + 216 = 0\]

Step 4: Intersection with the plane

The given plane equation is \(z = 5\). To find the intersection of the sphere with the plane, we can substitute this plane equation into the sphere equation: \[x^2 + y^2 + (5)^2 - 20x - 20y - 8(5) + 216=0\]

Step 5: Simplify the intersection equation

Now, let's simplify the equation: \[x^2 + y^2 + 25 - 20x - 20y - 40 + 216=0\] \[x^2 + y^2 - 20x - 20y + 201 = 0\] The equation describing the intersection of the sphere with the plane \(z=5\) is: \[x^2 + y^2 - 20x - 20y + 201 = 0\]

Key Concepts

Multivariable Calculus

In the realm of multivariable calculus, the understanding of three-dimensional space and surfaces is essential. Here, we delve into equations that describe not only lines and planes but also curves and surfaces in a three-dimensional setting. For instance, the equation of a sphere is a staple example of a surface that can be described in this space.

A sphere's equation, denoted by \( (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2 \), encapsulates the set of all points in three-dimensional space that are a fixed distance (the radius \( r \)) from a central point (the center \( (a, b, c) \)). Multivariable calculus uses such equations to explore geometric properties, calculate areas and volumes, and find intersection points or curves, which is particularly useful in the fields of physics, engineering, and computer graphics among others.

Additionally, concepts such as gradient, divergence and curl extend the possibilities of analysis by describing vector fields and the flow of particles throughout space. All these concepts are intertwined and crucial for a comprehensive grasp of the mathematical landscape that multivariable calculus unveils.

Sphere-Plane Intersection

When discussing sphere-plane intersection, we're looking at the curve of intersection where a plane slices through a sphere. This intersection can result in a circle, a point, or can be nonexistent (if, for instance, the plane 'misses' the sphere completely).

In our context, the sphere centered at \( (10, 10, 4) \) with a radius of 9 intersects with the plane \( z=5 \). To find this intersection, we substitute \( z=5 \) into the sphere's equation to yield an equation in terms of only \( x \) and \( y \) that represents a circle in the \( xy \) plane. The resulting equation, \( x^2 + y^2 - 20x - 20y + 201 = 0 \), walks us through the coordinates lying on the circular intersection.

This process is widely applicable in geometrical modeling, computer-aided design (CAD), and even in understanding planetary motion where satellite paths often intersect spheres representing celestial bodies.

Coordinate Geometry

The domain of coordinate geometry, also known as analytic geometry, serves as a bridge between algebra and geometry, providing algebraic answers to geometric questions. By placing geometric figures in a coordinated system, we can describe them through equations and analyze their properties algebraically.

For example, the center and radius of a sphere can be extracted from its algebraic equation. The equation \( x^2 + y^2 + z^2 - 20x - 20y - 8z + 216 = 0 \) can be rewritten in its standard form to reveal the sphere's center at \( (10, 10, 4) \) and its radius of 9. Coordinate geometry also enables us to determine distances, midpoints, and relations between figures, such as parallelism and perpendicularity.

Understanding this geometry is not only fundamental to succeeding in higher mathematics but is also essential in various technological endeavors. Whether in navigation systems, robotics, or architectural drafting, the principles of coordinate geometry lay the groundwork for precision and creativity in the sciences and beyond.