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Chapter 9: Problem 8

(a) Describe the set of points whose distance from the x-axis equals the distance from the yz-plane. \(\odot\) A cylinder opening along the \(x\) -axis \(\odot\) A cone opening along the x-axis \(\odot\) A cone opening along the z-axis \(\odot\) A cylinder opening along the z-axis \(\odot\) A cone opening along the y-axis \(\odot\) A cylinder opening along the y-axis (b) Find the equation for the set of points whose distance from the x-axis equals the distance from the yz-plane. \(\odot x^{2}+y^{2}=r^{2}\) \(\odot x^{2}+z^{2}=r^{2}\) \(\odot y^{2}=x^{2}+z^{2}\) (\odot) \(y^{2}+z^{2}=r^{2}\) \(\odot z^{2}=x^{2}+y^{2}\) \(\odot x^{2}=y^{2}+z^{2}\)

Short Answer

Expert verified
A set of points whose distance from the x-axis equals the distance from the yz-plane describes a cylinder opening along the x-axis. The equation for this set of points is \(x^2 = y^2 + z^2\).

Step by step solution

01

Understand the concept of distance from an axis or plane

The distance between a point (x, y, z) and the x-axis is given by its y and z coordinates' values. Hence, the distance between a point and the x-axis can be calculated as \( \sqrt{y^2+z^2}\). The distance between a point (x, y, z) and the yz-plane can be measured by the perpendicular distance from the point to the plane. As the yz-plane has the equation x = 0, the distance between a point and the yz-plane is given by the absolute value (or modulus) of the x-coordinate, i.e., |x|. We need to find the set of points and the equation for which the distances from the x-axis and yz-plane are equal.
02

Determine the set of points

We want to find the set of points (x, y, z) such that their distance from the x-axis is equal to their distance from the yz-plane. Mathematically, we can write this condition as: \( \sqrt{y^2+z^2} = |x| \) A point equidistant from x-axis and yz-plane indicates a surface that is symmetric about the x-axis with a specific distance along the x-axis at every point. Thus, the shape of this surface will be a cylinder opening along the x-axis. From the given options, the correct answer is: A cylinder opening along the x-axis.
03

Identify the equation for the given condition

We found the condition for the required set of points to be: \( \sqrt{y^2+z^2} = |x| \) To eliminate the square root and absolute value, we can square both sides of the equation: \( y^2+z^2 = x^2\) This equation represents a cylinder opening along the x-axis. From the given options, the correct answer is: \(x^2 = y^2 + z^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distance from an Axis
Understanding the distance from an axis in a 3D coordinate system is crucial for visualizing and solving problems involving geometric shapes in space. In the context of multivariable calculus, we often measure this distance as the perpendicular length from a point to an axis.

A point \textbf{(x, y, z)} has a distance to the x-axis determined solely by its y and z coordinates. Imagine a vertical line dropped from the point to the x-axis; the length of this line is the distance from the axis. Similarly, distances to the y-axis and z-axis are dependent on other coordinates. So, the distance from the point to the x-axis is given by the expression \( \text{distance to x-axis} = \text{sqrt}{\(y^2 + z^2\)} \).

This concept is a foundation for exploring the properties of various shapes and their relation to the coordinate axes.
Distance from a Plane
The distance from a plane, such as the yz-plane, is another concept that is essential in 3D geometry. For a point with coordinates \textbf{(x, y, z)}, this distance is the shortest line segment from the point to the plane.

Considering the yz-plane, which is defined by all points where x is zero (\(x = 0\)), the distance of any point \textbf{(x, y, z)} from this plane is simply the absolute value of the x-coordinate: \( \text{distance from yz-plane} = |x| \).

This concept helps in determining the position of points relative to planes in space and is fundamental to the study of projections and reflections in geometry.
3D Coordinate System
A 3D coordinate system is composed of three axes: x-axis, y-axis, and z-axis, typically assumed to be perpendicular to each other. This system allows us to uniquely identify the location of any point in a three-dimensional space by its coordinates \textbf{(x, y, z)}.

Each axis acts as a reference line from which distances are measured, and the origin (\(0, 0, 0\)) is where all three axes intersect. Problems in multivariable calculus often involve analyzing relationships between points and geometrical figures within this coordinate system.

By understanding how to work with this system, we form the basis for calculating distances, slopes, and various mathematical constructs essential for 3D spatial reasoning.
Cylindrical Surface Equation
The cylindrical surface equation in a 3D coordinate system defines the set of all points equidistant from a given axis, forming a cylindrical shape. A standard form of this equation, centered around the x-axis, looks like \(y^2 + z^2 = r^2\), where \(r\) is the radius of the cylinder.

This equation can be visualized as a collection of circles (with radius \(r\)) stacked along the x-axis. By setting the calculation of distances from the axis equal to distances from a plane, we derived the equation \(x^2 = y^2 + z^2\), which signifies a cylinder opening along the x-axis.

The understanding of this equation is crucial for students as it extends to more complex scenarios, including those involving rotations, volume integrals, and pressure in fluid dynamics.

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Most popular questions from this chapter

Perform the following operations on the vectors \(\vec{u}=\langle 0,5,-4\rangle, \vec{v}=\) \(\langle-2,0,3\rangle,\) and \(\vec{w}=\langle-3,0,1\rangle\). \(\vec{u} \cdot \vec{w}=\) _________ \((\vec{u} \cdot \vec{v}) \vec{u}=\)_________ \(((\vec{w} \cdot \vec{w}) \vec{u}) \cdot \vec{u}=\)_________ \(\vec{u} \cdot \vec{v}+\vec{v} \cdot \vec{w}=\)_________

For each surface, decide whether it could be a bowl, a plate, or neither. Consider a plate to be any fairly flat surface and a bowl to be anything that could hold water, assuming the positive z-axis is up. (a) \(x+y+z=3\) (b) \(z=1-x^{2}-y^{2}\) (c) \(z=x^{2}+y^{2}\) (d) \(z=-\sqrt{3-x^{2}-y^{2}}\) (e) \(z=1\)

What is the angle in radians between the vectors $$ \begin{array}{l} \mathbf{a}=(6,-4,9) \text { and } \\ \mathbf{b}=(5,-1,6) ? \end{array} $$ Angle:_______(radians)

The Ideal Gas Law, \(P V=R T\), relates the pressure \((P,\) in pascals), temperature \((T,\) in Kelvin \()\), and volume \((V,\) in cubic meters \()\) of 1 mole of a gas \(\left(R=8.314 \frac{\mathrm{J}}{\mathrm{mol} \mathrm{K}}\right.\) is the universal gas constant), and describes the behavior of gases that do not liquefy easily, such as oxygen and hydrogen. We can solve the ideal gas law for the volume and hence treat the volume as a function of the pressure and temperature: $$ V(P, T)=\frac{8.314 T}{P} $$ a. Explain in detail what the trace of \(V\) with \(P=1000\) tells us about a key relationship between two quantities. b. Explain in detail what the trace of \(V\) with \(T=5\) tells us. c. Explain in detail what the level curve \(V=0.5\) tells us. d. Use 2 or three additional traces in each direction to make a rough sketch of the surface over the domain of \(V\) where \(P\) and \(T\) are each nonnegative. Write at least one sentence that describes the way the surface looks. e. Based on all your work above, write a couple of sentences that describe the effects that temperature and pressure have on volume.

When people buy a large ticket item like a car or a house, they often take out a loan to make the purchase. The loan is paid back in monthly installments until the entire amount of the loan, plus interest, is paid. The monthly payment that the borrower has to make depends on the amount \(P\) of money borrowed (called the principal), the duration \(t\) of the loan in years, and the interest rate \(r\). For example, if we borrow $$\$ 18,000$$ to buy a car, the monthly payment \(M\) that we need to make to pay off the loan is given by the formula $$ M(r, t)=\frac{1500 r}{1-\frac{1}{\left(1+\frac{r}{12}\right)^{12 t}}} $$ a. Find the monthly payments on this loan if the interest rate is \(6 \%\) and the duration of the loan is 5 years. b. Create a table of values that illustrates the trace of \(M\) with \(r\) fixed at \(5 \% .\) Use yearly values of \(t\) from 2 to 6 . Round payments to the nearest penny. Explain in detail in words what this trace tells us about \(M\). c. Create a table of values that illustrates the trace of \(M\) with \(t\) fixed at 3 years. Use rates from \(3 \%\) to \(11 \%\) in increments of \(2 \%\). Round payments to the nearest penny. Explain in detail what this trace tells us about \(M\). d. Consider the combinations of interest rates and durations of loans that result in a monthly payment of $$\$ 200 .$$ Solve the equation \(M(r, t)=200\) for \(t\) to write the duration of the loan in terms of the interest rate. Graph this level curve and explain as best you can the relationship between \(t\) and \(r\).
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