Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ z=\sqrt{25-x^{2}-y^{2}} \quad c=0,1,2,3,4,5 $$

Step-by-Step Solution

Answer

The level curves of the function \(z=\sqrt{25-x^{2}-y^{2}}\) are given by \(y = ± \sqrt{25 - x^{2} - c^{2}}\) for the c values 0, 1, 2, 3, 4, 5. Each equation corresponds to a curve on the plane with restrictions for the x-variable given by \(25 - x^{2} - c^{2}\) >= 0

Step 1: Understanding Level Curves

Simply put, level curves of a function are the curves that result from the intersection of the function graph and a horizontal plane. They help provide a perception of the three-dimensional shape of the function.

Step 2: Setup Level Curve Equations

For each of the given c values, we set the function equal to c to get the level curve equations. This ultimately gives us: \(0 = \sqrt{25-x^{2}-y^{2}}\), \(1 = \sqrt{25-x^{2}-y^{2}}\), \(2 = \sqrt{25-x^{2}-y^{2}}\), \(3 = \sqrt{25-x^{2}-y^{2}}\), \(4 = \sqrt{25-x^{2}-y^{2}}\), and \(5 = \sqrt{25-x^{2}-y^{2}}\). After isolating x and y in each equation, we can obtain the level curves.

Step 3: Solve for x and y

Solving the equations for y by isolating y, we get: \(y^{2} = 25 - x^{2} - c^{2}\). Since we are to find the level curves, it is preferable to express y in terms of x for our sketch: \(y = ± \sqrt{25 - x^{2} - c^{2}}\) for \(c = 0, 1, 2, 3, 4, 5\). This gives six separate level curves.

Step 4: Sketching the curves

Sketching these curves out on a plane will help visualize them. For this it will also help to note that the domain of the square root must be >= 0, so \(25 - x^{2} - c^{2}\) >= 0. This will give additional restrictions for the x-variable for the level curves

Key Concepts

Graphing Functions

When graphing functions, especially involving level curves, it is crucial to understand how these curves graphically represent a function in two dimensions. A function like \(z = \sqrt{25 - x^{2} - y^{2}}\) forms a surface in three dimensions. The level curves come into play when we set \(z\) to specific values, which are also known as "c-values". This translates the problem into determining the curve where this slice of the surface happens at a particular height

• For each c-value, the equation forms a closed curve.
• The shape and the size of these curves can give insights into the nature and behavior of the function.

This approach aids greatly in visualizing and understanding the depth and layers of the surface through a series of these slices.

Equation Solving

Solving the equations corresponding to the given c-values helps us find the specific curves. For instance, if we set \(c = 2\), we’ll solve the equation \(2 = \sqrt{25 - x^{2} - y^{2}}\) to find the corresponding level curve.
The first step involves eliminating the square root by squaring both sides:

• \(c = \sqrt{25 - x^{2} - y^{2}} \Rightarrow c^{2} = 25 - x^{2} - y^{2}\)
• Simplify to express \(y\) in terms of \(x\): \(y^{2} = 25 - x^{2} - c^{2}\)

By solving these equations, one can compute the coordinates that satisfy this equation for specific \(c\) values. This procedure can then be repeated for each of the specified values of \(c\).

Domain Restrictions

Understanding domain restrictions is crucial for accurately sketching level curves.
The restriction arises because we’re dealing with the square root function, which requires non-negative inputs. For our problem, we must ensure:

• \(25 - x^{2} - y^{2} \geq 0\)
• This determines the range for which \(x\) and \(y\) values the function is defined.

For every \(c\), the relationship transforms the inequality to \(25 - x^{2} - c^{2} \geq 0\). This shows that as \(c\) increases, the allowable values for \(x\) decrease, tightening the domain.
This domain restriction directly affects the span of the graph, limiting \(x\) and \(y\) to a region defined by each \(c\). Understanding these restrictions helps prevent errors and highlights valid points for drawing.

Three-Dimensional Shapes

This exercise is a gateway into understanding three-dimensional shapes by studying their two-dimensional counterparts: the level curves. The original function \(z = \sqrt{25 - x^{2} - y^{2}}\) represents a sphere's upper half with a radius of 5 in three-dimensional space.
Level curves cut horizontally through this sphere at various heights:

• For different values of \(z = c\), the resulting level curves are circles shrinking toward the sphere's top.
• These circles become smaller as \(c\) approaches 5, finally reducing to a point when \(c = 5\).

This visualization helps one understand the physical arrangement of such shapes, reinforcing the idea of how surfaces in 3D space translate to 2D curves. Each level curve corresponds to a particular "slice" of the sphere, showcasing a fundamental interplay between dimensions.