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Chapter 15: Problem 1

Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point. \((2, \pi / 4,1) \quad\) (b) \((4,-\pi / 3,5)\)

Short Answer

Expert verified
(鈭�2, 鈭�2, 1) and (2, -2鈭�3, 5) are the rectangular coordinates.

Step by step solution

01

Understanding Cylindrical Coordinates

Cylindrical coordinates are given as \((r, \theta, z)\), where \(r\) is the radial distance, \(\theta\) is the angle in the xy-plane from the x-axis, and \(z\) is the height above the xy-plane. We have two points to consider here: \((2, \pi/4, 1)\) and \((4, -\pi/3, 5)\).
02

Converting to Rectangular Coordinates for (2, 蟺/4, 1)

To convert cylindrical coordinates \((r, \theta, z)\) to rectangular coordinates \((x, y, z)\), use the formulas: \(x = r \cos(\theta)\), \(y = r \sin(\theta)\), and \(z = z\).For the first point \((2, \pi/4, 1)\):- \(x = 2 \cos(\pi/4) = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2}\)- \(y = 2 \sin(\pi/4) = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2}\)- \(z = 1\)So, the rectangular coordinates are \((\sqrt{2}, \sqrt{2}, 1)\).
03

Converting to Rectangular Coordinates for (4, -蟺/3, 5)

Apply the same conversion formulas for the second point \((4, -\pi/3, 5)\):- \(x = 4 \cos(-\pi/3) = 4 \times \frac{1}{2} = 2\)- \(y = 4 \sin(-\pi/3) = 4 \times \left(-\frac{\sqrt{3}}{2}\right) = -2\sqrt{3}\)- \(z = 5\)Therefore, the rectangular coordinates are \((2, -2\sqrt{3}, 5)\).
04

Plotting the Points

To plot these points in three-dimensional space:1. For \((\sqrt{2}, \sqrt{2}, 1)\), locate a point in the xy-plane at \((\sqrt{2}, \sqrt{2})\) and then go up 1 unit along the z-axis.2. For \((2, -2\sqrt{3}, 5)\), locate a point in the xy-plane at \((2, -2\sqrt{3})\) and then go up 5 units along the z-axis.

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Key Concepts

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

Rectangular Coordinates
Rectangular coordinates are a way to describe a point in space using a grid-like system made up of three axes: the x-axis, y-axis, and z-axis. This system is also known as the Cartesian coordinate system. Each point is specified as \(x, y, z\), where \(x\) indicates the distance from the yz-plane, \(y\) indicates the distance from the xz-plane, and \(z\) indicates the height from the xy-plane.
Rectangular coordinates are commonly used because they align with our perception of space as a 3D grid, making them intuitive when working with three-dimensional objects.
  • For our example, converting cylindrical coordinates like \(2, \, \pi/4, \, 1\) gives rectangular coordinates \(\sqrt{2}, \, \sqrt{2}, \, 1\).
  • This means that the point is located \(\sqrt{2}\) units from both the yz and xz planes, and 1 unit above the xy-plane.
Coordinate System Conversion
Converting between coordinate systems is a crucial skill in mathematics and physics since different situations may require different models of describing space.
The process involves using specific mathematical formulas to translate between systems such as cylindrical and rectangular (Cartesian) coordinates. In the case of cylindrical to rectangular conversion:
  • The formula \(x = r \cos(\theta)\) calculates the x-coordinate using the radial distance \(r\) and angle \(\theta\).
  • The formula \(y = r \sin(\theta)\) calculates the y-coordinate.
  • The z-coordinate remains the same since both systems measure height from the same base plane.
These conversion steps ensure that the same physical point in space is accurately represented across different systems. For instance, using our coordinates \(2, \pi/4, 1\), this conversion yields \(\sqrt{2}, \sqrt{2}, 1\).
Graph Plotting
Graph plotting in three-dimensional space requires understanding the spatial representation of points after conversion from cylindrical to rectangular coordinates. By identifying points on a 3D coordinate system, you can visualize the exact location in space.
When plotting points such as \(\sqrt{2}, \sqrt{2}, 1\) or \(2, -2\sqrt{3}, 5\), follow these steps:
  • Start by locating the point in the xy-plane using the x and y values.
  • Once on the plane, move vertically up or down according to the z value.
This plotting method helps in creating clear visual representations of positions, which can simplify solving geometric problems or understanding physical phenomena.
Three-Dimensional Space
Understanding three-dimensional space is essential for dealing with complex real-world problems. Essentially, 3D space allows us to model and work with objects having width, depth, and height.
In three-dimensional space, points are represented as \(x,y,z\) coordinates. These coordinates define the location based on three intersecting planes. This spatial reasoning is crucial when dealing with problems in mechanics, architecture, and even computer graphics.
  • The \x\ and \y\ values indicate the position within a base plane, while \z\ specifies elevation or depth.
  • Visualizing these relationships can aid in tasks ranging from understanding geometric shapes to solving calculus problems about volumes and surfaces.
By plotting the given points from cylindrical coordinates after conversion, such as in \(\sqrt{2}, \sqrt{2}, 1\) and \(2, -2\sqrt{3}, 5\), you can see these mathematical concepts play out in concrete, visual ways.

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Most popular questions from this chapter

Use the Midpoint Rule for triple integrals (Exercise 24 to estimate the value of the integral. Divide \(B\) into eight sub-boxes of equal size. $$ \begin{array}{l}{\int \mathbb{N}_{B} \frac{1}{\ln (1+x+y+z)} d V, \text { where }} \\ {B=\\{(x, y, z) | 0 \leqslant x \leqslant 4,0 \leqslant y \leqslant 8,0 \leqslant z \leqslant 4\\}}\end{array} $$

(a) Evaluate \(\iiint_{E} d V,\) where \(E\) is the solid enclosed by the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1 .\) Use the transformation \(x=a u, y=b v, z=c w\) (b) The earth is not a perfect sphere; rotation has resulted in flattening at the poles. So the shape can be approximated by an ellipsoid with \(a=b=6378 \mathrm{km}\) and \(c=6356 \mathrm{km} .\) Use part (a) to estimate the volume of the earth.

The average value of a function \(f(x, y, z)\) over a solid region \(E\) is defined to be $$ f_{\mathrm{ave}}=\frac{1}{V E)} \iiint_{E} f(x, y, z) d V $$ where \(V(E)\) is the volume of \(E .\) For instance, if \(\rho\) is a density function, then \(\rho_{\text { ave is the average density of } E .}\) Find the average value of the function \(f(x, y, z)=x^{2} z+y^{2} z\) over the region enclosed by the paraboloid \(z=1-x^{2}-y^{2}\) and the plane \(z=0 .\)

(a) A lamp has two bulbs of a type with an average lifetime of 1000 hours. Assuming that we can model the probability of failure of these bulbs by an exponential density function with mean \(\mu=1000,\) find the probontity that both of the lamp's bulbs fail within 1000 hours. (b) Another lamp has just one bulb of the same type as in part (a). If one bulb burns out and is replaced by a bulb of the same type, find the probability that the two bulbs fail within a total of 1000 hours.

Evaluate the integral by reversing the order of integration. $$\int_{0}^{\sqrt{\pi}} \int_{y}^{\sqrt{\pi}} \cos \left(x^{2}\right) d x d y$$
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