Problem 19
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
Verified Answer
The graph is the upper hemisphere of a sphere centered at the origin with radius 4.
1Step 1: Recognize the equation type
The given equation is in the form of \( z = \sqrt{16 - x^2 - y^2} \). This is the equation of the upper hemisphere of a sphere. The square root implies it is only the upper half.
2Step 2: Identify the sphere's properties
The equation \( x^2 + y^2 + z^2 = 16 \) represents a sphere centered at the origin with a radius of 4. The given equation is half of this sphere, where the non-negative square root indicates the upper hemisphere where \( z \geq 0 \).
3Step 3: Sketch the graph
To sketch the graph, draw a three-dimensional coordinate system. Plot a sphere centered at the origin with a radius of 4. Shade only the upper hemisphere (above the xy-plane) since only \( z = \sqrt{16 - x^2 - y^2} \) is considered, meaning \( z \geq 0 \).
4Step 4: Analyze the domain and range
In the equation, \( 16 - x^2 - y^2 \geq 0 \), so the domain is all points \((x, y)\) within the circle \( x^2 + y^2 \leq 16 \) in the xy-plane. The range of \( z \) is \( [0, 4] \), covering the upper hemisphere.
Key Concepts
Spheres and HemispheresCoordinate SystemsDomain and Range
Spheres and Hemispheres
The concept of spheres and hemispheres is fundamental in understanding three-dimensional geometry. A **sphere** is a perfectly symmetrical object in three dimensions, where every point on the surface is equidistant from a specified central point. Mathematically, it's defined by the equation \[ x^2 + y^2 + z^2 = r^2 \]where \( r \) is the radius of the sphere. All points satisfying this equation form a spherical surface.
On the other hand, a **hemisphere** refers to either the top or bottom half of a sphere, divided by a plane passing through its center. When talking about the upper hemisphere, we usually describe it by the square root function, like in our exercise: \[ z = \sqrt{r^2 - x^2 - y^2} \]
Only the points where \( z \geq 0 \) are considered, hence forming the upper half only. This property is essential when visualizing or modeling real-world problems that involve spherical shapes, like the earth's surface or domes.
On the other hand, a **hemisphere** refers to either the top or bottom half of a sphere, divided by a plane passing through its center. When talking about the upper hemisphere, we usually describe it by the square root function, like in our exercise: \[ z = \sqrt{r^2 - x^2 - y^2} \]
Only the points where \( z \geq 0 \) are considered, hence forming the upper half only. This property is essential when visualizing or modeling real-world problems that involve spherical shapes, like the earth's surface or domes.
Coordinate Systems
A coordinate system in three-dimensional space extends the familiar x and y axes into the third dimension with the z-axis. This addition allows us to locate points and sketch equations that define surfaces or shapes in a more complex spatial context.
In a 3D Cartesian coordinate system:
By visualizing these axes and how coordinates (x, y, z) intersect in space, you'll better grasp abstract mathematical surfaces like spheres or hemispheres within this coordinate framework.
In a 3D Cartesian coordinate system:
- The **x-axis** runs horizontally.
- The **y-axis** extends perpendicularly to the x-axis on a flat plane.
- The **z-axis** stands vertically, perpendicular to both the x and y axes.
By visualizing these axes and how coordinates (x, y, z) intersect in space, you'll better grasp abstract mathematical surfaces like spheres or hemispheres within this coordinate framework.
Domain and Range
In mathematics, the terms **domain** and **range** refer to the set of possible inputs and outputs of a function, respectively. In the given equation \[ z = \sqrt{16 - x^2 - y^2} \], understanding these concepts helps determine which values are valid.
- The **domain** is all possible \[ (x, y) \] pairs that result in a valid \( z \) value. This means evaluating \[ 16 - x^2 - y^2 \geq 0 \] to ensure the expression under the square root is non-negative. This condition forms a circle of radius 4 in the xy-plane, defined by \[ x^2 + y^2 \leq 16 \].
- The **range** of the function relates to permissible output values for \( z \), given the constraint that the square root results are only non-negative. For our exercise, this means \( z \) values range from 0 to 4.
Comprehending the domain and range assists in accurately sketching functions and comprehending the behavior of different mathematical surfaces, such as detecting the dimensions and extent of a hemisphere in this particular graph.
- The **domain** is all possible \[ (x, y) \] pairs that result in a valid \( z \) value. This means evaluating \[ 16 - x^2 - y^2 \geq 0 \] to ensure the expression under the square root is non-negative. This condition forms a circle of radius 4 in the xy-plane, defined by \[ x^2 + y^2 \leq 16 \].
- The **range** of the function relates to permissible output values for \( z \), given the constraint that the square root results are only non-negative. For our exercise, this means \( z \) values range from 0 to 4.
Comprehending the domain and range assists in accurately sketching functions and comprehending the behavior of different mathematical surfaces, such as detecting the dimensions and extent of a hemisphere in this particular graph.
Other exercises in this chapter
Problem 19
Sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, \(y^{2}-4 x^{2}=20,(2,6)\)
View solution Problem 19
For what numbers \(c\) are \(\langle c, 6\rangle\) and \(\langle c,-4\rangle\) orthogonal?
View solution Problem 19
Find the equation of the plane containing the line \(x=1+2 t, y=-1+3 t, z=4+t\) and the point \((1,-1,5)\).
View solution Problem 20
Find the equation of the plane through the origin that is perpendicular to the \(x y\)-plane and the plane \(3 x-2 y+z=4\).
View solution