Problem 35 - Calculus an Applied Approach

Question

Find the standard equation of the sphere. $$ \text { Center: }(1,1,5) ; \text { radius: } 3 $$

Step-by-Step Solution

Answer

The standard equation of the sphere with center at (1,1,5) and radius 3 is \((x-1)^2 + (y-1)^2 + (z-5)^2 = 9\).

Step 1: Identify the Components of the Equation

Given that the center of the sphere is at (1,1,5) and radius \(r\) is equal to 3. The formula for the standard equation of a sphere is \((x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\). In this equation (a,b,c) stand for the coordinates of the center and \(r\) for the radius of the sphere.

Step 2: Substitute Given Values Into the Standard Equation

Substitute a=1, b=1, c=5 and \(r\) =3 into the formula. Doing this we get, \((x-1)^2 + (y-1)^2 + (z-5)^2 = 3^2\)

Step 3: Simplify the Equation

The equation becomes \((x-1)^2 + (y-1)^2 + (z-5)^2 = 9\).

Key Concepts

Sphere

A sphere is a perfectly round three-dimensional shape, much like a ball or globe. Its defining characteristic is that every point on its surface is equidistant from its center. This property makes it one of the most symmetrical objects in geometry.

In mathematics, spheres are important shapes, and they can be described using equations. The standard equation of a sphere helps us define it in a coordinate system, providing a way to calculate positions and distances related to the sphere. This equation reflects the balance and symmetry inherent in its shape.

• The formula \((x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\) represents the sphere in a three-dimensional space.
• Every point \(x, y, z\) that satisfies this equation falls on the sphere's surface.
• The center \((a, b, c)\) and radius \(r\) are crucial in defining the exact size and position of the sphere.

Radius

The radius of a sphere is the distance from its center to any point on its surface. This is a constant measure because all points on the surface are the same distance from the center. The radius is half the diameter of the sphere and is a key factor in calculating many of its properties.

In the standard sphere equation, the radius \(r\) appears squared on the right side of the equation as \(r^2\). This is because in geometry, distances are often squared—calculating squared deviations is a fundamental part of many geometric relationships.

• Mathematicians use the radius to calculate the sphere's surface area, using the formula \(4\pi r^2\).
• It's also used to find the volume of the sphere, calculated by \(\frac{4}{3}\pi r^3\).
• In our example, the sphere's radius is 3, making its role critical in determining the size and extent of the sphere.

Center of a Sphere

The center of a sphere is a point in space that is exactly in the middle of the sphere. It acts like the heart of the sphere, where if you could insert lines from the surface, they would all meet at this central point.

In the standard equation of a sphere \((x-a)^2 + (y-b)^2 + (z-c)^2 = r^2\), the coordinates \((a, b, c)\) specify the center of the sphere. Each of these values adjusts the position of the sphere along the respective axis in a 3D coordinate system.

• The center helps define the sphere's location in space relative to other objects.
• Changing the center coordinates simply moves the sphere without altering its size or shape.
• In our example, the center is located at \(1, 1, 5\), positioning the sphere uniquely within the coordinate system.