Problem 55 - Calculus Early Transcendentals

Question

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the \(x y-, x z^{-}\), and \(y z\) -traces, when they exist. c. Sketch a graph of the surface. $$\frac{x^{2}}{25}+\frac{y^{2}}{9}-z^{2}=1$$

Step-by-Step Solution

Answer

Answer: The graph represents a hyperboloid of two sheets. Question: What are the x-intercepts of the graph? Answer: There are two x-intercepts at (-5,0,0) and (5,0,0). Question: What are the y-intercepts of the graph? Answer: There are two y-intercepts at (0,-3,0) and (0,3,0). Question: Are there any z-intercepts? Answer: No, there are no z-intercepts. Question: What is the equation of the xy-trace? Answer: The equation of the xy-trace is $\frac{x^2}{25}+\frac{y^2}{9}=1$, which is an ellipse. Question: What is the equation of the xz-trace? Answer: The equation of the xz-trace is $\frac{x^2}{25}-z^2=1$, which is a hyperbola. Question: What is the equation of the yz-trace? Answer: The equation of the yz-trace is $\frac{y^2}{9}-z^2=1$, which is another hyperbola.

Step 1: X-axis Intercept

To find the x-intercept, set y=0 and z=0 and solve for x: $$\frac{x^2}{25}+\frac{(0)^2}{9}-(0)^2=1$$ $$\frac{x^2}{25}=1$$ Now, solve for x. #Step 2: Calculate the x-intercept#

Step 2: X-axis Intercept

Multiply both sides by 25 and take the square root: $$x^2=25$$ $$x=\pm\sqrt{25}$$ $$x=-5,5$$ There are two x-intercepts at \((-5,0,0)\) and \((5,0,0)\). #Step 3: Set two coordinates to 0 and solve for the other one#

Step 3: Y-axis Intercept

To find the y-intercept, set x=0 and z=0 and solve for y: $$\frac{(0)^2}{25}+\frac{y^2}{9}-(0)^2=1$$ $$\frac{y^2}{9}=1$$ Now, solve for y. #Step 4: Calculate the y-intercept#

Step 4: Y-axis Intercept

Multiply both sides by 9 and take the square root: $$y^2=9$$ $$y=\pm\sqrt{9}$$ $$y=-3,3$$ There are two y-intercepts at \((0,-3,0)\) and \((0,3,0)\). #Step 5: Set two coordinates to 0 and solve for the other one#

Step 5: Z-axis Intercept

To find the z-intercept, set x=0 and y=0 and solve for z: $$\frac{(0)^2}{25}+\frac{(0)^2}{9}-z^2=1$$ $$-z^2=1$$ Since there is no real solution for z, there are no z-intercepts. #b. Find the equations of the xy-, xz-, and yz-traces# #Step 6: Find xy-trace equation#

Step 6: XY-Trace Equation

Set z=0 and get the resulting equation: $$\frac{x^2}{25}+\frac{y^2}{9}=1$$ This is the equation of an ellipse. #Step 7: Find xz-trace equation#

Step 7: XZ-Trace Equation

Set y=0 and get the resulting equation: $$\frac{x^2}{25}-z^2=1$$ This is the equation of a hyperbola. #Step 8: Find yz-trace equation#

Step 8: YZ-Trace Equation

Set x=0 and get the resulting equation: $$\frac{y^2}{9}-z^2=1$$ This is another equation of a hyperbola. #c. Sketch a graph of the surface#

Step 9: Graph the Surface

Based on the information we've found, we can now sketch a graph of the surface. The graph is a hyperboloid of two sheets, with the major axis along the x-axis. In the xy-plane, we see an ellipse. Along the xz- and yz-planes, we see hyperbolas. The x-intercepts are at \((-5,0,0)\) and \((5,0,0)\), while the y-intercepts are at \((0,-3,0)\) and \((0,3,0)\). There are no z-intercepts.

Key Concepts

Intercepts

Intercepts are points where a graph crosses the axes. In this context, we look for where the given quadric surface crosses the x, y, and z axes. To find each intercept, we set the other two variables to zero and solve for the remaining one.

• X-Intercepts: Set y=0 and z=0; solve \( \frac{x^2}{25} = 1 \). We find \( x = \pm 5 \), resulting in intercepts at (-5,0,0) and (5,0,0).
• Y-Intercepts: Set x=0 and z=0; solve \( \frac{y^2}{9} = 1 \). We find \( y = \pm 3 \), giving intercepts at (0,-3,0) and (0,3,0).
• Z-Intercepts: Set x=0 and y=0; \( -z^2 = 1 \) leads to no real solution, so there are no z-intercepts.

Finding intercepts is a fundamental skill for understanding the orientation and position of a surface in space.

Traces

Traces are the cross-sectional shapes you get when you slice a solid with a plane. In our equation of the quadric surface, traces are found by setting one variable to zero and solving the resulting equations.

• XY-Trace: Set \( z = 0 \) and solve \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \), showing an ellipse. This trace helps to understand the elliptic nature of the surface in the xy-plane.
• XZ-Trace: Set \( y = 0 \) and solve \( \frac{x^2}{25} - z^2 = 1 \), resulting in a hyperbola. This indicates a hyperbolic form across the xz-plane.
• YZ-Trace: Set \( x = 0 \) and solve \( \frac{y^2}{9} - z^2 = 1 \), which is also a hyperbola. Like the xz-trace, this shows a hyperbolic curve along the yz-plane.

These traces are essential for accurately sketching the surface and understanding its geometric properties.

Hyperboloid of Two Sheets

A hyperboloid of two sheets is a unique type of surface found in 3D geometry. It arises from the negative term being isolated in an equation of a quadric surface. With the given equation: \[ \frac{x^2}{25} + \frac{y^2}{9} - z^2 = 1 \] we observe this structure because of the \(-z^2\). Here's why this shape is significant:

• Unlike its one-sheet cousin, this type forms two separated surfaces, or 'sheets'.
• The major axis runs parallel to the axis with the positive terms in the equation – here, it's the x-axis.
• The absence of z-intercepts hints at its disconnected character along the z-axis.

This surface is often compared to a pair of bowls opening in opposite directions, exemplifying its stunning symmetry and distinctive shape.