/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Consider a rectangular box with ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 1

Consider a rectangular box with one of the vertices at the origin, as shown in the following figure. If point \(A(2,3,5)\) is the opposite vertex to the origin, then find a. the coordinates of the other six vertices of the box and b. the length of the diagonal of the box determined by the vertices \(O\) and \(A\).

Short Answer

Expert verified
The other six vertices are (0,3,5), (2,0,5), (2,3,0), (0,0,5), (0,3,0), (2,0,0), and the diagonal length is \(\sqrt{38}\).

Step by step solution

01

Understanding the Box Setup

The box is a 3D rectangular shape where one vertex is at the origin, er O = (0,0,0), and the opposite vertex, A, is at (2,3,5). These coordinates mean the box spans 2 units along the x-axis, 3 units along the y-axis, and 5 units along the z-axis.
02

Listing Other Vertices

Each face of the box is parallel to one of the coordinate planes, hence, the other 6 vertices can be written by varying the respective components: 1. (0,3,5) - corresponds to shifts along y and z from origin, 2. (2,0,5) - corresponds to shifts along x and z from origin, 3. (2,3,0) - corresponds to shifts along x and y from origin, 4. (0,0,5) - shift along z only from origin, 5. (0,3,0) - shift along y only from origin, 6. (2,0,0) - shift along x only from origin.
03

Finding the Diagonal Length

The length of the diagonal of the box connecting points O (at the origin) and A (at 2,3,5) is calculated using the distance formula in 3D: \[\sqrt{(2-0)^2 + (3-0)^2 + (5-0)^2}\]Simplifying that gives: \[ \sqrt{2^2 + 3^2 + 5^2} = \sqrt{4 + 9 + 25} = \sqrt{38} \] Thus, the length of the diagonal is \( \sqrt{38} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

3D Coordinates
Three-dimensional (or 3D) coordinates help us determine specific locations in space. In 3D geometry, each coordinate is expressed as a triplet \((x, y, z)\). These values represent distances from three perpendicular axes:
  • x-axis: Usually runs horizontally.
  • y-axis: Typically extends vertically upwards.
  • z-axis: Points in or out of the page/screen, depending on orientation.
For example, a coordinate \((2, 3, 5)\) can be broken down as:
  • 2 units in the positive x direction
  • 3 units in the positive y direction
  • 5 units in the positive z direction
This kind of representation allows us to identify and work with points in 3D space, a key feature in solving geometry-related problems.
Vertices of a Rectangular Box
In a 3D space, a rectangular box consists of vertices鈥攕pecific points that define its shape and dimensions. If one vertex is at the origin \((0, 0, 0)\), and the opposite vertex \(A\) is at \((2, 3, 5)\), we can easily identify the rest based on shifts along the axes.

Identifying the Remaining Vertices

Let鈥檚 determine the other six vertices:
  • (0,3,5): Move along y and z axes from the origin.
  • (2,0,5): Shift along x and z axes from the origin.
  • (2,3,0): Adjust along x and y axes from the origin.
  • (0,0,5): Shift along the z axis.
  • (0,3,0): Move along the y axis.
  • (2,0,0): Adjust only along the x axis.
These vertices help visualize the box, illustrating its extent in 3D space.
Distance Formula in 3D
Understanding how to calculate the distance between two points in 3D space is crucial for tasks like finding the length of a diagonal across a rectangular box.

Applying the Distance Formula

To find the diagonal from the origin to a point \((x, y, z)\), use the distance formula:\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]Here, if one point is at the origin \((0,0,0)\) and the other is at \((2,3,5)\), the calculation is straightforward:
  • Subtract each coordinate from the point at the origin: \((2-0), (3-0), (5-0)\)
  • Square each difference: \(2^2, 3^2, 5^2\)
  • Add the squares: \(4 + 9 + 25 = 38\)
  • Take the square root: \(\sqrt{38}\)
This shows that the diagonal length is \(\sqrt{38}\), enabling a better understanding of box dimensions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For the following exercises, find the equation of the plane with the given properties.The plane that passes through points \((0,1,5),(2,-1,6)\), and \((3,2,5)\).

In cartography, Earth is approximated by an oblate spheroid rather than a sphere. The radii at the equator and poles are approximately \(3963 \mathrm{mi}\) and \(3950 \mathrm{mi}\), respectively. a. Write the equation in standard form of the ellipsoid that represents the shape of Earth. Assume the center of Earth is at the origin and that the trace formed by plane \(z=0\) corresponds to the equator. b. Sketch the graph. c. Find the equation of the intersection curve of the surface with plane \(z=1000\) that is parallel to the xy-plane. The intersection curve is called a parallel. d. Find the equation of the intersection curve of the surface with plane \(x+y=0\) that passes through the \(z\) -axis. The intersection curve is called a meridian.

Write the standard form of the equation of the ellipsoid centered at the origin that passes through points \(A(2,0,0), B(0,0,1)\), and \(C\left(\frac{1}{2}, \sqrt{11}, \frac{1}{2}\right)\)

A heart-shaped surface is given by equation \(\left(x^{2}+\frac{9}{4} y^{2}+z^{2}-1\right)^{3}-x^{2} z^{3}-\frac{9}{80} y^{2} z^{3}=0\). a. Use a CAS to graph the surface that models this shape. b. Determine and sketch the trace of the heart-shaped surface on the \(x z\) -plane.

Two children are playing with a ball. The girl throws the ball to the boy. The ball travels in the air, curves \(3 \mathrm{ft}\) to the right, and falls \(5 \mathrm{ft}\) away from the girl (see the following figure). If the plane that contains the trajectory of the ball is perpendicular to the ground, find its equation.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.