Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name and sketch the graph of each of the following equations in three-space. $$ z=\sqrt{16-x^{2}-y^{2}} $$

Step-by-Step Solution

Answer

The graph is the upper hemisphere of a sphere with radius 4 centered at the origin.

Step 1: Identify the Type of Equation

The given equation is \( z = \sqrt{16-x^2-y^2} \). Recognize that this is a standard form for the upper hemisphere of a sphere centered at the origin with radius 4 because the equation fits the form \( z = \sqrt{R^2 - x^2 - y^2} \).

Step 2: Determine the Radius

Compare the equation to the form \( z = \sqrt{R^2 - x^2 - y^2} \). Here, \( R^2 = 16 \), so the radius \( R \) is \( \sqrt{16} = 4 \).

Step 3: Interpret the Domain of the Equation

Since \( z = \sqrt{16-x^2-y^2} \), the quantity inside the square root must be non-negative, which imposes the restriction \( 16-x^2-y^2 \geq 0 \). This describes a region where \( x^2 + y^2 \leq 16 \), which is the area under a circle of radius 4 in the xy-plane.

Step 4: Sketch the Graph

The graph represents the upper hemisphere of a sphere. Draw the xy-plane, and sketch a circle with radius 4. The surface rises from \( z = 0 \) at the edge of the circle to \( z = 4 \) directly above the center of the circle (the origin). The hemisphere is symmetric about the z-axis.

Step 5: Label the Graph

Label the maximum height at \( z = 4 \) above the origin and the intersection with the xy-plane as a circle of radius 4. This is a key characteristic of the hemisphere.

Key Concepts

Spherical Coordinates

Spherical coordinates are a fantastic tool for understanding three-dimensional space. Instead of relying solely on rectangular coordinates \(x, y, z\), spherical coordinates allow us to define the position of a point using three values: the radius \(\rho\), the polar angle \(\theta\), and the azimuthal angle \(\phi\). This system is especially useful in understanding shapes like spheres.

• \(\rho\): The distance from the origin to the point. It describes how far a point is from the center of the coordinate system.
• \(\theta\): The angle in the xy-plane starting from the x-axis. This helps identify the point's direction in the horizontal plane.
• \(\phi\): The angle from the z-axis downwards. It indicates the height of the point measured from the z-axis down.

Using these definitions, any point in 3D can be described effectively. Considering our exercise, converting a sphere's equation from rectangular to spherical coordinates can be insightful. For a sphere of radius \(4\) centered at the origin, spherical coordinates succinctly express any point on the sphere's surface.

Equation of a Sphere

The equation of a sphere in three-dimensional space is a simple yet powerful tool. The general form is \(x^2 + y^2 + z^2 = R^2\), where \(R\) represents the sphere's radius. If the sphere is centered at the origin, this form makes visualizing the three-dimensional object much simpler.
In our exercise, the equation \(z = \sqrt{16 - x^2 - y^2}\) closely relates to the equation of a sphere. By squaring both sides, we can recognize the relationship: \(x^2 + y^2 + z^2 = 16\). This equation describes a sphere with a radius of \(4\) and tells us that the sphere is centered at the origin. However, the presence of the square root function in the given equation limits the graph to the upper hemisphere.
Understanding this concept allows us to effectively differentiate between portions of the sphere and comprehend how they interact with the coordinate axes.

Mathematical Visualization

Mathematical visualization is essential for comprehending complex three-dimensional shapes like spheres. It enables us to translate abstract equations into tangible images, making it easier to grasp their meaning. The graph given in the exercise represents the upper hemisphere of a sphere, which can be identified by its equation \(z = \sqrt{16 - x^2 - y^2}\).
When visualizing this, consider the following:

• A circle on the xy-plane with a radius of 4 represents the base of the hemisphere.
• From this base, the surface rises smoothly to its maximum height at \(z = 4\), which is directly above the origin.
• The surface is perfectly symmetrical around the z-axis, highlighting the spherical nature of the shape.

By sketching these elements clearly, you can easily represent the shape and understand how the components of the sphere fit together in three-dimensional space.