Problem 64 - University Calculus: Early Transcendentals

Question

Find equations for the spheres whose centers and radii are given $$ \begin{array}{ll} \text { Center } & \text { Radius } \\ (0,-7,0) & 7 \end{array} $$

Step-by-Step Solution

Answer

The sphere's equation is \(x^2 + (y + 7)^2 + z^2 = 49\).

Step 1: Understand the equation of a sphere

The general equation for a sphere with center \((h, k, l)\) and radius \(r\) is given by \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\). This formula represents the set of all points \((x, y, z)\) that are equidistant (distance = radius) from the center.

Step 2: Apply the center coordinates

We are given the center of the sphere as \((0, -7, 0)\). Substitute these values into the center coordinates \((h, k, l)\) in the sphere's equation. So, \(h = 0\), \(k = -7\), and \(l = 0\). This changes the equation to \((x - 0)^2 + (y + 7)^2 + (z - 0)^2 = r^2\).

Step 3: Substitute the radius value

The given radius is 7. Substitute \(r = 7\) into the equation. This gives us \((x - 0)^2 + (y + 7)^2 + (z - 0)^2 = 49\).

Step 4: Simplify the equation

Since subtracting or adding zero does not change the value, we can simplify the expression to \(x^2 + (y + 7)^2 + z^2 = 49\). This is the equation of the sphere with the specified center and radius.

Key Concepts

Center coordinates

Understanding the center coordinates of a sphere is crucial for expressing its equation. In three-dimensional geometry, the center of a sphere is defined by a point with coordinates \(h, k, l\), representing the position in the \(x\)-axis, \(y\)-axis, and \(z\)-axis respectively.
Visualizing the center is like pinpointing the exact location of the sphere in a 3D space.

• The center is crucial because it acts as the reference point for measuring the radius, which is the distance from the center to any point on the surface of the sphere.
• For example, in our exercise, the center coordinates are \(0, -7, 0\). Here, the center is positioned right on the z-axis since the x and z coordinates are zero.
• These coordinates affect only how the sphere is positioned within the space, not its shape or size.

This position is a significant aspect used in calculating the sphere equation and understanding its placement in 3D geometry.

Radius

The radius of a sphere plays a fundamental role in shaping its equation. It is the constant value representing the distance from the center of the sphere to any point on its surface.
The radius is always a positive number, and it determines the size of the sphere.

• In mathematical terms, each point on the sphere's surface is equidistant from the center, making the radius uniform all around.
• In our example, the radius is given as 7. This means every point that forms the sphere is 7 units away from the center located at \(0, -7, 0\).
• This constant distance, the radius, is squared in the equation formula, appearing as \(r^2\). In our case, \(7^2 = 49\).

Understanding the radius helps in realizing how large the sphere is and how it extends in the three-dimensional space around its center.

General equation of a sphere

The general equation of a sphere is a mathematical representation that connects the center coordinates and the radius together.
This equation expresses all locations that form the entire surface of the sphere in a 3D space.

• The form of this equation is \((x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\), where \(h, k, l\) are the center coordinates and \(r\) is the radius.
• Each squared term corresponds to the distance of a point from the center along its respective axis.
• This equation essentially captures the essence of what it means to be a sphere—every point on its surface is the same distance from the center.
• In our specific instance, replacing \(h, k, l\) with \(0, -7, 0\) and \(r\) with 7, we derive the equation \(x^2 + (y + 7)^2 + z^2 = 49\).

The general equation provides a comprehensive view of how spheres are structured in geometry, illustrating both their size and location with elegance and efficiency.